91Ó°ÊÓ

Show that the number of solutions in nonnegative integers of the inequality $$x_{1}+x_{2}+\cdots+x_{n} \leq M$$ where \(M\) is a nonnegative integer, is \(C(M+n, n)\).

How many bridge deals are there? (A deal consists of partitioning a 52 -card deck into four hands, each containing 13 cards.)

Suppose there are 10 roads from \(\mathrm{Oz}\) to Mid Earth and five roads from Mid Earth to Fantasy Island. How many round-trips are there of the form Oz-Mid EarthFantasy Island-Mid Earth-Oz in which on the return trip we do not reverse the original route from \(\mathrm{Oz}\) to Fantasy Island?

Show that $$\sum_{k=m}^{n} C(k, m) H_{k}=C(n+1, m+1)\left(H_{n+1}-\frac{1}{m+1}\right)$$for all \(n \geq m,\) where \(H_{k},\) the \(k\) th harmonic number, is defined$$ H_{k}=\sum_{i=1}^{k} \frac{1}{i} $$

In how many ways can three teams containing four, two, and two persons be selected from a group of eight persons?

If the coin is flipped 10 times, what is the probability of no heads?

A domino is a rectangle divided into two squares with each square numbered one of \(0,1, \ldots, 6,\) repetitions allowed. How many distinct dominoes are there?

How many eight-bit strings contain exactly three 0 's?

If the coin is flipped 10 times, what is the probability of at least one head?

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