91Ó°ÊÓ

Prove that if \(p\) is prime, \(C(p, i)\) is divisible by \(p\) for all \(i\) \(1 \leq i \leq p-1\)

How many eight-bit strings have exactly two 1's?

How many eight-bit strings contain at least two 0 's in a row?

How many eight-bit strings have at least one \(1 ?\)

Find the number of (unordered) five-card poker hands, selected from an ordinary 52 -card deck, having the properties indicated. Containing four aces

How many eight-bit strings read the same from either end? (An example of such an eight-bit string is \(01111110 .\) Such strings are called palindromes.)

Prove that a planar polygon with \(n\) sides, \(n \geq 3,\) has at least three interior angles each less than 180 degrees. Assume no 0-degree interior angles. As an example, in the following figure angles \(A, C,\) and \(E\) are each less than 180 degrees.

A six-person committee composed of Alice, Ben, Connie, Dolph, Egbert, and Francisco is to select a chairperson, secretary, and treasurer: How many selections are there in which both Ben and Francisco are officers?

Find the number of (unordered) five-card poker hands, selected from an ordinary 52 -card deck, having the properties indicated. Containing cards of exactly two suits

In how many ways can we place 10 identical balls in 12 boxes if each box can hold one ball?