91Ó°ÊÓ

How many bit strings of length 12 contain a) exactly three 1s? b) at most three 1s? c) at least three 1s? d) an equal number of 0s and 1s?

Use the binomial theorem to find the coefficient of \(x^{a} y^{b}\) in the expansion of \(\left(5 x^{2}+2 y^{3}\right)^{6}\) , where a) \(a=6, b=9\) b) \(a=2, b=15\) c) \(a=3, b=12\) d) \(a=12, b=0\) e) \(a=8, b=9\)

How many bit strings are there of length six or less, not counting the empty string?

Let \(\left(x_{i}, y_{i}\right), i=1,2,3,4,5,\) be a set of five distinct points with integer coordinates in the \(x y\) plane. Show that the midpoint of the line joining at least one pair of these points has integer coordinates.

A book publisher has 3000 copies of a discrete mathematics book. How many ways are there to store these books in their three warehouses if the copies of the book are indistinguishable?

Let \(\left(x_{i}, y_{i}, z_{i}\right), i=1,2,3,4,5,6,7,8,9,\) be a set of nine distinct points with integer coordinates in \(x y z\) space. Show that the midpoint of at least one pair of these points has integer coordinates.

How many bit strings with length not exceeding n, where n is a positive integer, consist entirely of 1s, not counting the empty string?

Use the binomial theorem to find the coefficient of \(x^{a} y^{b}\) in the expansion of \(\left(2 x^{3}-4 y^{2}\right)^{7}\) , where a) \(a=9, b=8\) b) \(a=8, b=9\) c) \(a=0, b=14\) d \(a=12, b=6\) e) \(a=18, b=2\)

How many bit strings of length n, where n is a positive integer, start and end with 1s?

How many ordered pairs of integers \((a, b)\) are needed to guarantee that there are two ordered pairs \(\left(a_{1}, b_{1}\right)\) and \(\left(a_{2}, b_{2}\right)\) such that \(a_{1} \bmod 5=a_{2} \bmod 5\) and \(b_{1} \bmod 5=b_{2} \bmod 5 ?\)