14. In this problem you will prove some results about the binomial coefficients, using induction. Recall...
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14. In this problem you will prove some results about the binomial coefficients, using induction. Recall that () - n! (n – k)!k!' where n is a positive integer, and 0 <k < n. (a) Prove that n - 1 n 2 2 and k < n. Hint: You do not need induction to prove this. Bear in mind that 0! = 1. (b) Verify that () by induction on n that () is an integer, for all k with 0 <k < n. (Note: You may have encountered (") as the count of the number of k element subsets of a set of n objects; it follows from this that () is an integer. What we are asking for here is an inductive proof based on algebra.) (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, 1 and (") = 1. Use these facts, together with part (a), to prove %3D (x + v)" = E()- n-k %3D k=0 14. In this problem you will prove some results about the binomial coefficients, using induction. Recall that () - n! (n – k)!k!' where n is a positive integer, and 0 <k < n. (a) Prove that n - 1 n 2 2 and k < n. Hint: You do not need induction to prove this. Bear in mind that 0! = 1. (b) Verify that () by induction on n that () is an integer, for all k with 0 <k < n. (Note: You may have encountered (") as the count of the number of k element subsets of a set of n objects; it follows from this that () is an integer. What we are asking for here is an inductive proof based on algebra.) (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, 1 and (") = 1. Use these facts, together with part (a), to prove %3D (x + v)" = E()- n-k %3D k=0
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