answer:To find the number of distinct k-dimensional subspaces in the vector space Fp^n, we can utilize the concept of combinations of subspaces and the properties of finite fields. First, let's consider how to construct a k-dimensional subspace. We can start by choosing k linearly independent vectors from the n-dimensional vector space Fp^n. The first vector can be chosen in p^n - 1 ways (since there are p^n vectors in total, and we exclude the zero vector). Once we have chosen the first vector, the second vector must be linearly independent of the first one. There are p^n - p ways to choose the second vector, because there are p multiples of the first vector that we need to exclude. Following this pattern, when choosing the i-th vector (for 1 ≤ i ≤ k), there are p^n - p^(i-1) ways to choose it, as we need to exclude all linear combinations of the previously chosen i-1 vectors. However, we are not interested in the order in which we choose these k vectors. Therefore, we need to account for the fact that there are (p^k - 1)(p^k - p)...(p^k - p^(k-1)) distinct ordered bases for a given k-dimensional subspace. Now, to find the total number of distinct k-dimensional subspaces, we can multiply the number of ways to choose k linearly independent vectors and divide by the number of distinct ordered bases for each subspace: [(p^n - 1)(p^n - p)...(p^n - p^(k-1))] / [(p^k - 1)(p^k - p)...(p^k - p^(k-1))] Simplifying this expression, we get: (p^n - 1)(p^(n-1) - 1)...(p^(n-k+1) - 1) / (p^k - 1)(p^(k-1) - 1)...(p - 1) This is the number of distinct k-dimensional subspaces in the vector space Fp^n. It's worth noting that this result can also be expressed using the Gaussian binomial coefficient, denoted as [n choose k]_p: [n choose k]_p = (p^n - 1)(p^(n-1) - 1)...(p^(n-k+1) - 1) / (p^k - 1)(p^(k-1) - 1)...(p - 1) The Gaussian binomial coefficient plays a crucial role in various combinatorial problems involving finite fields and vector spaces, and it has several interesting properties and applications.

answer:To solve this problem, let's consider the rates at which each hose fills the pool. We'll denote the rate of hose A as A, hose B as B, and hose C as C. The volume of the pool is irrelevant as long as it's constant, so we'll consider the rates and time in terms of pool volumes per hour. From the problem statement, we have the following information: - Hoses A and B together fill the pool in 4 hours, so their combined rate is 1/4 pool per hour: A + B = 1/4. - Hoses A and C together fill the pool in 5 hours, so their combined rate is 1/5 pool per hour: A + C = 1/5. - Hoses B and C together fill the pool in 6 hours, so their combined rate is 1/6 pool per hour: B + C = 1/6. We're asked to find how long it takes for all three hoses to fill the pool together, which means we need to find their combined rate A + B + C. Adding the three equations together, we get: 2(A + B + C) = 1/4 + 1/5 + 1/6. To add the fractions on the right side, we find the least common denominator, which is 60. Then we can rewrite each fraction: 2(A + B + C) = 15/60 + 12/60 + 10/60 = 37/60. Divide both sides by 2: A + B + C = 37/120. This is the combined rate of all three hoses. Since rate and time are inversely proportional, we can find the time it takes for all three hoses to fill the pool by taking the reciprocal of their combined rate: time = 1 / (37/120) = 120 / 37 hours. So, it takes approximately 3.24 hours (120/37) for all three hoses A, B, and C working together to fill the pool.

answer:To solve this problem, we'll follow the same approach as before, considering the rates at which each hose fills the pool. We'll denote the rate of hose A as A, hose B as B, and hose C as C. From the problem statement, we have the following information: - Hoses A and B together fill the pool in 2.5 hours, so their combined rate is 1/2.5 pools per hour: A + B = 1/2.5 = 2/5. - Hoses A and C together fill the pool in 3 hours, so their combined rate is 1/3 pool per hour: A + C = 1/3. - Hoses B and C together fill the pool in 3.6 hours, so their combined rate is 1/3.6 pools per hour: B + C = 1/3.6 = 5/18. We're asked to find how long it takes for all three hoses to fill the pool together, which means we need to find their combined rate A + B + C. Adding the three equations together, we get: 2(A + B + C) = 2/5 + 1/3 + 5/18. To add the fractions on the right side, we find the least common denominator, which is 90. Then we can rewrite each fraction: 2(A + B + C) = 36/90 + 30/90 + 25/90 = 91/90. Divide both sides by 2: A + B + C = 91/180. This is the combined rate of all three hoses. Since rate and time are inversely proportional, we can find the time it takes for all three hoses to fill the pool by taking the reciprocal of their combined rate: time = 1 / (91/180) = 180 / 91 hours. So, it takes approximately 1.98 hours (180/91) for all three hoses A, B, and C working together to fill the pool.

answer:To find the time it takes for each hose to fill the pool individually, we need to find the rates A, B, and C, and then take the reciprocal of each rate. From the previous problem, we have the following equations: - A + B = 2/5 - A + C = 1/3 - B + C = 5/18 We can rewrite these equations as: - A = 2/5 - B - A = 1/3 - C - B = 5/18 - C Since the left-hand sides of the first two equations are equal, we can set them equal to each other: 2/5 - B = 1/3 - C Rearranging the equation, we get: C - B = 1/3 - 2/5 Finding the least common denominator (15), we can rewrite the fractions: C - B = 5/15 - 6/15 = -1/15 Now, we can rewrite this equation as: C = B - 1/15 Substituting this expression for C into the equation B + C = 5/18, we get: B + B - 1/15 = 5/18 Combine like terms: 2B - 1/15 = 5/18 Add 1/15 to both sides: 2B = 5/18 + 1/15 Find the least common denominator (90): 2B = 25/90 + 6/90 = 31/90 Divide both sides by 2: B = 31/180 Now that we have B, we can find C: C = B - 1/15 = 31/180 - 12/180 = 19/180 Now we have B and C, we can find A using one of the original equations: A + B = 2/5 A = 2/5 - B A = 2/5 - 31/180 Find the least common denominator (180): A = 72/180 - 31/180 A = 41/180 Now we have the rates A, B, and C. To find the time it takes for each hose to fill the pool individually, we take the reciprocal of each rate: Time for A = 1 / (41/180) = 180 / 41 ≈ 4.39 hours Time for B = 1 / (31/180) = 180 / 31 ≈ 5.81 hours Time for C = 1 / (19/180) = 180 / 19 ≈ 9.47 hours So, it takes approximately 4.39 hours for hose A, 5.81 hours for hose B, and 9.47 hours for hose C to fill the pool individually.