14. In this problem you will prove some results about the binomial coefficients, using induction. Recall...

 

Transcribed Image Text:

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. 1 and (") = 1. Use these facts, together with part (a), to prove (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.) %3D (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, n (x + y)" = E()- n-k 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. 1 and (") = 1. Use these facts, together with part (a), to prove (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.) %3D (c) Use part (a) and induction to prove the Binomial Theorem: For non-negative n and variables x, y, n (x + y)" = E()- n-k k=0

Answer rating: 100% (QA)

The answer... View the full answer