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Show that if six integers are chosen at random, then at least two of them will have the same remainder when divided by 5 .

Write out Row 11 of Pascal's triangle.

Suppose a set \(A\) has 37 elements. How many subsets of \(A\) have 10 elements? How many subsets have 30 elements? How many have 0 elements?

Five cards are dealt off of a standard 52 -card deck and lined up in a row. How many such lineups are there that have at least one red card? How many such lineups are there in which the cards are either all black or all hearts?

Use the binomial theorem to find the coefficient of \(x^{8} y^{5}\) in \((x+y)^{13}\).

Airports are identified with 3 -letter codes. For example, Richmond, Virginia has the code \(R I C\), and Memphis, Tennessee has \(M E M\). How many different 3 -letter codes are possible?

How many 4 -digit positive integers are there for which there are no repeated digits, or for which there may be repeated digits, but all digits are odd?

You deal a pile of cards, face down, from a standard 52 -card deck. What is the least number of cards the pile must have before you can be assured that it contains at least five cards of the same suit?

How many 2 -element multisets can be made from the 26 letters of the alphabet?

Five cards are dealt off of a standard 52 -card deck and lined up in a row. How many such lineups are there in which all 5 cards are of the same suit?