1 Combinations a Prove and provide an intuitive explanation

1. Combinations (a) Prove and provide an intuitive explanation for the following identities: (b) Binomial Expansion: Show that for any real numbers a, b we have: Conclude that (c) What is the number of all possible subsets of a set with n elements? (d) Pascal\'s triangle: Show that

Solution

a)

C(n, r) = C(n, n-r)

When you take r at a time, the remaining n - r was like \"taken by another person\". Everytime you take r, the n - r that person has becomes different as well, so you have the same number of ways.

C(n, n) = 1

There\'s only one way to take all items, that is, to get them all.

C(n, 1) = n

There are n ways to get 1 at a time, that is, because there are n items there.

*******************

Hi! Please submit the next part as a separate question. That way we can continue helping you! Please indicate which parts are not yet solved when you submit. Thanks!