Chapter 9.6, Problem 62E

To determine

Find the total number of ways of a player selecting six numbers

Answer to Problem 62E

The total number of ways of a player selecting six numbers is ${}_{40}{C}_6=3838380$

Explanation of Solution

Given:

Consider a game in which a player chooses six distinct numbers from 1 to 40. That means the total numbers are 40 and the player selects 6 out of the 40 numbers. Here 6 numbers are selected without considering an order so use combination to find the number of ways.

In general, the number of combinations of $n$ elements taken $r$ at a time is,

  ${}_n{C}_r=\frac{n!}{(n-r)!r!}$

Here,

  $\begin{array}{l}n=40\hfill \\ r=6\hfill \end{array}$

Put the values in the formula and solve as,

  $\begin{array}{cc}{}_{40}{C}_6=& \frac{40!}{(40-6)!6!}\\ & =\frac{40!}{6!34!}\\ & =\frac{40\times 39\times 38\times 37\times 36\times 35\times 34!}{6!\times 34!}\\ & =\frac{2763633600}{720}\\ & =3838380\end{array}$

Hence, the total number of ways of a player selecting six numbers is ${}_{40}{C}_6=3838380$