answer：Okay, so I have this problem where I need to distribute 6 volunteers into 4 different groups, and each group has to have at least one person. These groups are going to serve at four different pavilions of the World Expo. I need to figure out how many different distribution plans there are. Hmm, let's break this down.First, I know that distributing volunteers into groups can be thought of in terms of functions in mathematics. Each volunteer can be assigned to any of the four groups, right? So, for each volunteer, there are 4 choices. Since there are 6 volunteers, the total number of ways to assign them without any restrictions would be 4 multiplied by itself 6 times, which is 4^6. Let me calculate that: 4^6 is 4096. So, there are 4096 possible ways if I don't care about having empty groups.But wait, the problem says each group must have at least one person. That means I can't have any group left empty. So, I need to subtract the cases where one or more groups are empty. This seems like a classic inclusion-exclusion problem.Let me recall how inclusion-exclusion works. If I have a total number of possibilities and I want to exclude certain cases, I subtract the cases where one condition is violated, then add back the cases where two conditions are violated, subtract the cases where three conditions are violated, and so on.In this case, the "conditions" are that each group must have at least one person. So, the first step is to subtract the distributions where at least one group is empty. How many ways are there where at least one group is empty?Well, if one specific group is empty, then all 6 volunteers must be distributed among the remaining 3 groups. The number of ways to do this is 3^6. But since there are 4 groups, any of the 4 could be the one that's empty. So, I need to multiply 3^6 by 4. Let me compute 3^6: that's 729. So, 4 times 729 is 2916. Therefore, I need to subtract 2916 from the total.But wait, if I subtract 2916, I might be over-subtracting. Because when two groups are empty, those cases were subtracted twice. So, I need to add them back once. How many ways are there where two specific groups are empty? If two groups are empty, all volunteers must go into the remaining 2 groups. The number of ways is 2^6. There are 4 groups, so the number of ways to choose which two groups are empty is C(4,2), which is 6. So, 6 times 2^6 is 6 times 64, which is 384. Therefore, I need to add 384 back.Now, what about if three groups are empty? That would mean all volunteers are in one group. The number of ways is 1^6, which is 1. There are C(4,3) = 4 ways to choose which three groups are empty. So, 4 times 1 is 4. But since we're dealing with inclusion-exclusion, after subtracting and adding, we might have to subtract this again. However, in our case, since we're only required to have at least one person in each group, having three groups empty is actually not allowed, so we might not need to consider this. But to be thorough, let's see.In inclusion-exclusion, we alternate between subtracting and adding. So, after subtracting the single empty groups, adding the double empty groups, we would subtract the triple empty groups. But in this problem, triple empty groups are invalid because we need each group to have at least one person. So, perhaps we don't need to go that far. But let's check.If I subtract 4^6 - 4*3^6 + 6*2^6 - 4*1^6, that would be 4096 - 2916 + 384 - 4. Let's compute that:4096 - 2916 is 1180.1180 + 384 is 1564.1564 - 4 is 1560.Wait, but earlier I thought we didn't need to subtract the triple empty groups because they are invalid. So, which is correct?Actually, in inclusion-exclusion, we do consider all the terms up to the number of sets, which in this case is 4. So, we should include the term for three empty groups. But let me think again.The formula for the number of onto functions from a set A to a set B is given by:|B|^{|A|} - C(|B|,1)*(|B|-1)^{|A|} + C(|B|,2)*(|B|-2)^{|A|} - ... + (-1)^{|B|}*C(|B|,|B|)*(|B|-|B|)^{|A|}In this case, |A| is 6 (volunteers) and |B| is 4 (groups). So, the formula becomes:4^6 - C(4,1)*3^6 + C(4,2)*2^6 - C(4,3)*1^6 + C(4,4)*0^6But 0^6 is 0, so the last term is 0. Therefore, the formula simplifies to:4^6 - 4*3^6 + 6*2^6 - 4*1^6Which is 4096 - 2916 + 384 - 4 = 1560.Wait, but earlier I thought the answer was 1564. So, which one is correct?Let me double-check the calculations:4^6 = 40963^6 = 729, so 4*729 = 29162^6 = 64, so 6*64 = 3841^6 = 1, so 4*1 = 4So, 4096 - 2916 = 11801180 + 384 = 15641564 - 4 = 1560So, the correct total is 1560.But wait, earlier I thought that triple empty groups are invalid, but according to the inclusion-exclusion principle, we do need to subtract them because they were overcounted in the previous steps.So, the correct number is 1560.But let me think again. Is this the number of onto functions? Yes, because we're distributing volunteers into groups with each group having at least one person, which is equivalent to onto functions from the set of volunteers to the set of groups.Alternatively, another way to think about this is using Stirling numbers of the second kind, which count the number of ways to partition a set of n objects into k non-empty subsets. The formula is S(n,k), and then multiplied by k! if the groups are distinguishable.In this case, since the groups are different pavilions, they are distinguishable. So, the number of ways is S(6,4) * 4!.What is S(6,4)? Let me recall the formula for Stirling numbers of the second kind:S(n,k) = S(n-1,k-1) + k*S(n-1,k)With base cases S(n,1) = 1 and S(n,n) = 1.So, let's compute S(6,4):We can build up the Stirling numbers:S(1,1) = 1S(2,1) = 1, S(2,2) = 1S(3,1) = 1, S(3,2) = 3, S(3,3) = 1S(4,1) = 1, S(4,2) = 7, S(4,3) = 6, S(4,4) = 1S(5,1) = 1, S(5,2) = 15, S(5,3) = 25, S(5,4) = 10, S(5,5) = 1S(6,1) = 1, S(6,2) = 31, S(6,3) = 90, S(6,4) = 65, S(6,5) = 15, S(6,6) = 1Wait, so S(6,4) is 65.Then, the number of ways is 65 * 4! = 65 * 24 = 1560.So, that matches the inclusion-exclusion result.Therefore, the number of different distribution plans is 1560.But wait, earlier I thought it was 1564, but that was before considering the triple empty groups. So, the correct answer is 1560.But let me double-check the inclusion-exclusion calculation:4^6 = 40964*3^6 = 4*729 = 29166*2^6 = 6*64 = 3844*1^6 = 4*1 = 4So, 4096 - 2916 = 11801180 + 384 = 15641564 - 4 = 1560Yes, that's correct.Alternatively, using Stirling numbers: S(6,4) = 65, 65 * 24 = 1560.So, both methods give the same result.Therefore, the number of different distribution plans is 1560.

answer：First, I need to convert the repeating decimals (0.overline{36}) and (0.overline{09}) into fractions.For (0.overline{36}), since the repeating part has two digits, I can set it equal to (frac{36}{99}).Similarly, for (0.overline{09}), the repeating part also has two digits, so it equals (frac{9}{99}).Next, I divide the two fractions:[frac{frac{36}{99}}{frac{9}{99}} = frac{36}{99} times frac{99}{9} = frac{36}{9} = 4]Therefore, the result of the division is 4.

answer：Alright, so I've got this probability problem here about rolling two dice, one red and one green. Each die has numbers from 1 to 6, and they're both fair, meaning each number has an equal chance of coming up. The problem defines three events:- **M**: The sum of the two dice is 7.- **N**: The product of the two dice is odd.- **G**: The number on the red die is greater than 3.And then there are four statements (A, B, C, D) about whether these events are mutually exclusive, mutually exhaustive, or independent. I need to figure out which of these statements are correct.First, let me make sure I understand the terms:- **Mutually exclusive**: Two events are mutually exclusive if they cannot happen at the same time. In other words, there's no overlap between their outcomes. - **Mutually exhaustive**: Two events are mutually exhaustive if together they cover all possible outcomes. That means every possible outcome must be in at least one of the two events.- **Independent**: Two events are independent if the occurrence of one doesn't affect the probability of the other. Mathematically, this means that the probability of both events happening together is equal to the product of their individual probabilities.Okay, so I need to analyze each pair of events (M and N, M and G, N and G) and check if they're mutually exclusive, mutually exhaustive, or independent.Let's start by listing out all possible outcomes when rolling two dice. Each die has 6 faces, so there are 6 x 6 = 36 possible outcomes. Each outcome is a pair (x, y) where x is the result of the red die and y is the result of the green die.Now, let's define each event more concretely.**Event M (x + y = 7):**To find all outcomes where the sum is 7, I can list them:- (1, 6)- (2, 5)- (3, 4)- (4, 3)- (5, 2)- (6, 1)So, there are 6 outcomes in event M.**Event N (xy is odd):**For the product of two numbers to be odd, both numbers must be odd. So, x and y both have to be odd numbers. The odd numbers on a die are 1, 3, and 5.So, the possible outcomes for event N are all pairs where both x and y are odd:- (1, 1)- (1, 3)- (1, 5)- (3, 1)- (3, 3)- (3, 5)- (5, 1)- (5, 3)- (5, 5)That's 9 outcomes in total for event N.**Event G (x > 3):**This event is only concerned with the red die. It doesn't matter what the green die shows; we just need the red die to show a number greater than 3. So, x can be 4, 5, or 6.For each possible value of x (4, 5, 6), y can be any number from 1 to 6. So, the number of outcomes in event G is 3 x 6 = 18.Alright, now that I have all the events defined, let's tackle each statement one by one.**Statement A: M and N are mutually exclusive but not mutually exhaustive.**First, are M and N mutually exclusive? That means, is there any outcome that is in both M and N?Looking at event M: All outcomes where the sum is 7.Looking at event N: All outcomes where both numbers are odd.So, for an outcome to be in both M and N, it needs to have a sum of 7 and both numbers must be odd.Let's see if any of the outcomes in M have both numbers odd.Looking at the outcomes in M:- (1, 6): 1 is odd, 6 is even. Not both odd.- (2, 5): 2 is even, 5 is odd. Not both odd.- (3, 4): 3 is odd, 4 is even. Not both odd.- (4, 3): 4 is even, 3 is odd. Not both odd.- (5, 2): 5 is odd, 2 is even. Not both odd.- (6, 1): 6 is even, 1 is odd. Not both odd.So, none of the outcomes in M are in N. Therefore, M and N are mutually exclusive.Now, are they mutually exhaustive? That would mean that every possible outcome is either in M or in N.But clearly, there are outcomes that are neither in M nor in N. For example, (2, 2) is neither a sum of 7 nor a product of odd numbers. Similarly, (4, 4) is neither in M nor in N.Therefore, M and N are mutually exclusive but not mutually exhaustive. So, statement A is correct.**Statement B: M and N are mutually exhaustive.**We just saw that M and N are not mutually exhaustive because there are outcomes that are neither in M nor in N. So, statement B is incorrect.**Statement C: M and G are mutually independent.**To check if M and G are independent, we need to see if the probability of both M and G occurring together is equal to the product of their individual probabilities.First, let's find P(M):There are 6 outcomes in M out of 36 possible, so P(M) = 6/36 = 1/6.Next, P(G):There are 18 outcomes in G out of 36 possible, so P(G) = 18/36 = 1/2.Now, let's find P(M and G). This is the probability that both M and G occur, meaning the sum is 7 and the red die is greater than 3.Looking at the outcomes in M:- (1, 6): x = 1, which is not greater than 3.- (2, 5): x = 2, which is not greater than 3.- (3, 4): x = 3, which is not greater than 3.- (4, 3): x = 4, which is greater than 3.- (5, 2): x = 5, which is greater than 3.- (6, 1): x = 6, which is greater than 3.So, out of the 6 outcomes in M, 3 of them have x > 3. Therefore, P(M and G) = 3/36 = 1/12.Now, let's check if P(M and G) equals P(M) * P(G):P(M) * P(G) = (1/6) * (1/2) = 1/12.Since P(M and G) = 1/12 and P(M) * P(G) = 1/12, they are equal. Therefore, M and G are independent. So, statement C is correct.**Statement D: N and G are mutually independent.**Again, we need to check if P(N and G) equals P(N) * P(G).First, let's find P(N):There are 9 outcomes in N out of 36 possible, so P(N) = 9/36 = 1/4.We already know P(G) = 1/2.Now, let's find P(N and G). This is the probability that both N and G occur, meaning the product is odd and the red die is greater than 3.Looking at event N, which requires both x and y to be odd. Additionally, x must be greater than 3.So, x can be 5 (since 4 is even and 3 is not greater than 3). Wait, x has to be greater than 3 and odd. So, x can be 5.Wait, hold on. x can be 4, 5, or 6 for G, but for N, x has to be odd. So, x can only be 5 in this case.Similarly, y has to be odd: 1, 3, or 5.So, the outcomes where both N and G occur are:- (5, 1)- (5, 3)- (5, 5)That's 3 outcomes.Therefore, P(N and G) = 3/36 = 1/12.Now, let's check if this equals P(N) * P(G):P(N) * P(G) = (1/4) * (1/2) = 1/8.But P(N and G) = 1/12, which is not equal to 1/8. Therefore, N and G are not independent. So, statement D is incorrect.**Summary:**- A: Correct- B: Incorrect- C: Correct- D: IncorrectSo, the correct conclusions are A and C.

answer：Okay, so I have this geometry problem here. Let me try to visualize it first. There's a point P and a line ℓ. The distance from P to ℓ is 12 units. Then there's a point T on ℓ such that the distance from P to T is 13 units. I need to find the radius of a circle that passes through P and is tangent to ℓ at T.Alright, let me break this down. First, since the distance from P to ℓ is 12, that means if I drop a perpendicular from P to ℓ, the length of that perpendicular is 12. Let me denote the foot of this perpendicular as point X. So, PX = 12.Now, point T is on ℓ, and PT = 13. So, triangle PXT is a right triangle with legs PX = 12 and XT, and hypotenuse PT = 13. I can use the Pythagorean theorem to find XT. Let me calculate that:XT² + PX² = PT² XT² + 12² = 13² XT² + 144 = 169 XT² = 169 - 144 XT² = 25 XT = 5So, XT is 5 units. That means the distance from X to T along the line ℓ is 5 units.Now, I need to find the radius of the circle that passes through P and is tangent to ℓ at T. Let me denote the center of this circle as O. Since the circle is tangent to ℓ at T, the radius OT must be perpendicular to ℓ. Therefore, OT is perpendicular to ℓ, which means OT is equal to the distance from O to ℓ, which is the radius r.Also, since the circle passes through P, the distance from O to P must also be equal to the radius r. So, OP = OT = r.Let me try to find the coordinates of these points to make it easier. Let's set up a coordinate system where the line ℓ is the x-axis. Then, point X, the foot of the perpendicular from P to ℓ, will be at (a, 0) for some a. Point P will then be at (a, 12). Point T is on ℓ, so it will be at (t, 0). We know that PT = 13, so the distance between (a, 12) and (t, 0) is 13.Using the distance formula:√[(t - a)² + (0 - 12)²] = 13 √[(t - a)² + 144] = 13 (t - a)² + 144 = 169 (t - a)² = 25 t - a = ±5So, t = a + 5 or t = a - 5. Therefore, the distance between X and T is 5 units, which matches what I found earlier.Now, let's find the coordinates of the center O. Since the circle is tangent to ℓ at T, the center O must lie directly above T on the line perpendicular to ℓ at T. Since ℓ is the x-axis, this perpendicular is the vertical line x = t. So, the coordinates of O are (t, r), where r is the radius.We also know that the distance from O to P must be equal to r. So, the distance between (t, r) and (a, 12) is r. Let's write that equation:√[(t - a)² + (r - 12)²] = rBut we already know that (t - a)² = 25 from earlier. So, substituting that in:√[25 + (r - 12)²] = rLet me square both sides to eliminate the square root:25 + (r - 12)² = r² 25 + r² - 24r + 144 = r² 25 + 144 - 24r + r² = r² 169 - 24r = 0 -24r = -169 r = 169 / 24So, the radius r is 169/24. Let me check if that makes sense. 169 divided by 24 is approximately 7.04, which seems reasonable given the distances involved.Wait, let me double-check my steps. I set up the coordinate system with ℓ as the x-axis, found the coordinates of P and T, then determined the center O must be at (t, r). Then, using the distance from O to P equals r, I set up the equation and solved for r. Everything seems to check out.I think that's the correct answer. The radius of the circle is 169/24 units.**Final Answer**The radius of the circle is boxed{dfrac{169}{24}}.

answer：Okay, so I have this problem about cat food packages. There are large and small packages, and a large one has more food than a small one. The problem says that one large package and four small packages are enough to feed a cat for exactly two weeks, which is 14 days. Now, the question is asking if one large package and three small packages are necessarily enough to feed the cat for 11 days.Alright, let's break this down. First, I need to figure out how much food is in each package. Let me assign some variables to make this easier. Let's say L is the amount of food in a large package, and S is the amount of food in a small package. The problem tells me that one large and four small packages last for 14 days. So, I can write that as:L + 4S = 14 daysNow, the question is asking if one large and three small packages will last for 11 days. So, I need to check if:L + 3S ≥ 11 daysBut wait, I don't know the exact values of L and S. I just know their relationship from the first equation. Maybe I can express L in terms of S or vice versa. Let's try that.From the first equation:L = 14 - 4SSo, if I substitute this into the second equation, I get:(14 - 4S) + 3S ≥ 11Simplify that:14 - S ≥ 11Subtract 14 from both sides:-S ≥ -3Multiply both sides by -1 (and reverse the inequality sign):S ≤ 3Hmm, so this tells me that each small package must be enough to feed the cat for 3 days or less. But wait, does that make sense? If a small package is only 3 days, then four small packages would be 12 days, and adding a large package would make it 14 days. So, the large package would be 2 days worth of food. But the problem says a large package has more food than a small one. If a small package is 3 days, the large package should be more than 3 days, but according to this, it's only 2 days. That doesn't make sense.Maybe I made a mistake in my reasoning. Let me check again.I have:L + 4S = 14And I want to find if L + 3S ≥ 11From the first equation, L = 14 - 4SSubstitute into the second:14 - 4S + 3S ≥ 11Which simplifies to:14 - S ≥ 11Then:-S ≥ -3Multiply by -1:S ≤ 3So, S must be less than or equal to 3 days. But if S is 3 days, then L would be 14 - 4*3 = 14 - 12 = 2 days. But L is supposed to be larger than S, which would mean L > S. If S is 3, L would have to be more than 3, but according to this, L is only 2, which contradicts the given information.So, maybe S has to be less than 3 days. Let's say S is 2 days. Then L would be 14 - 4*2 = 14 - 8 = 6 days. That makes sense because L is larger than S. Now, let's check if L + 3S is enough for 11 days.L + 3S = 6 + 3*2 = 6 + 6 = 12 days12 days is more than 11 days, so yes, it would be enough.Wait, but what if S is less than 2 days? Let's say S is 1 day. Then L would be 14 - 4*1 = 10 days. Then L + 3S = 10 + 3*1 = 13 days, which is still more than 11 days.So, regardless of the value of S, as long as L is larger than S, L + 3S will always be more than 11 days. Because if S is 3, L is 2, but that contradicts L being larger than S. So, S must be less than 3, making L larger, and thus L + 3S will definitely be more than 11 days.Therefore, one large package and three small packages are necessarily enough to feed the cat for 11 days.

answer：Okay, so I have this trigonometry problem here. It says that cosα is 3/5, and α is between -π/2 and 0. I need to find the value of sin2α divided by (1 - cos2α). The options are A: -4/3, B: 4/3, C: -3/4, D: 3/4. Hmm, let me think about how to approach this.First, I remember that cosα is given as 3/5, and α is in the fourth quadrant because it's between -π/2 and 0. In the fourth quadrant, cosine is positive, which matches the given value, and sine is negative. So, I can find sinα using the Pythagorean identity.The Pythagorean identity is sin²α + cos²α = 1. Since cosα is 3/5, cos²α is (3/5)² = 9/25. So, sin²α = 1 - 9/25 = 16/25. Taking the square root, sinα is either 4/5 or -4/5. But since α is in the fourth quadrant, sinα is negative. Therefore, sinα is -4/5.Now, I need to find sin2α and cos2α. I remember the double-angle formulas: sin2α = 2sinαcosα and cos2α = 2cos²α - 1. Let me compute these.First, sin2α = 2 * (-4/5) * (3/5) = 2 * (-12/25) = -24/25.Next, cos2α = 2*(3/5)² - 1 = 2*(9/25) - 1 = 18/25 - 1 = 18/25 - 25/25 = -7/25.Now, I need to compute sin2α divided by (1 - cos2α). Let's write that out:sin2α / (1 - cos2α) = (-24/25) / (1 - (-7/25)).First, simplify the denominator: 1 - (-7/25) = 1 + 7/25 = 32/25.So now, the expression becomes (-24/25) / (32/25). Dividing two fractions is the same as multiplying by the reciprocal, so:(-24/25) * (25/32) = (-24 * 25) / (25 * 32).I notice that 25 cancels out from numerator and denominator, so it simplifies to (-24)/32.Simplify this fraction by dividing numerator and denominator by 8: (-24 ÷ 8)/(32 ÷ 8) = (-3)/4.So, the value of sin2α / (1 - cos2α) is -3/4.Looking back at the options, that's option C.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. Found sinα correctly using Pythagorean identity, considering the quadrant.2. Applied double-angle formulas correctly for sin2α and cos2α.3. Calculated sin2α as -24/25 and cos2α as -7/25.4. Substituted into the expression, simplified the denominator correctly to 32/25.5. Divided the fractions correctly, canceled out the 25s, simplified to -24/32, then reduced to -3/4.Everything seems correct. So, yes, the answer should be C: -3/4.**Final Answer**boxed{C}