91Ó°ÊÓ

Show that among any group of five (not necessarily consecutive) integers, there are two with the same remainder when divided by 4.

Let \(d\) be a positive integer. Show that among any group of \(d+1\) (not necessarily consecutive) integers there are two with exactly the same remainder when they are divided by \(d .\)

What is the coefficient of \(x^{8} y^{9}\) in the expansion of \((3 x+2 y)^{17} ?\)

How many different three-letter initials are there that begin with an A?

How many possibilities are there for the win, place, and show (first, second, and third) positions in a horse race with 12 horses if all orders of finish are possible?

A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernickel bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are there to choose a) six bagels? b) a dozen bagels? c) two dozen bagels? d) a dozen bagels with at least one of each kind? e) a dozen bagels with at least three egg bagels and no more than two salty bagels?

Let \(n\) be a positive integer. Show that in any set of \(n\) consecutive integers there is exactly one divisible by \(n .\)

A croissant shop has plain croissants, cherry croissants, chocolate croissants, almond croissants, apple croissants, and broccoli croissants. How many ways are there to choose a) a dozen croissants? b) three dozen croissants? c) two dozen croissants with at least two of each kind? d) two dozen croissants with no more than two broccoli croissants? e) two dozen croissants with at least five chocolate croissants and at least three almond croissants? f ) two dozen croissants with at least one plain croissant, at least two cherry croissants, at least three chocolate croissants, at least one almond croissant, at least two apple croissants, and no more than three broccoli croissants?

How many bit strings of length 10 contain a) exactly four 1s? b) at most four 1s? c) at least four 1s? d) an equal number of 0s and 1s?

How many ways are there to choose eight coins from a piggy bank containing 100 identical pennies and 80 identical nickels?