Chapter 13.2, Problem 30AYU

List all the combinationss of 6 objects 1, 2, 3, 4, 5, and 6 taken 3 at a time. What is $C\left(6,3\right)$?

To determine

To write:

a. All the combinations of 6 objects 1, 2, 3, 4, 5 and 6 taken 3 at a time.

b. What is $C\left(6,3\right)?$

Answer to Problem 30AYU

a. $\left\{123, 124, 125, 126, 134,135,136, 145, 146,156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456\right\}$

b. 20

Explanation of Solution

Given:

The 6 objects 1, 2, 3, 4, 5 and 6 taken 3 at a time.

Formula used:

$C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Calculation:

The list of ordered arrangements of 6 objects 1, 2, 3, 4, 5 and 6 choosing 3 at a time $\left\{123, 124, 125, 126, 134,135,136, 145, 146,156, 234, 235, 236, 245, 246, 256, 345, 346, 356, 456\right\}$.

$C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

$C\left(6,3\right)=\frac{6!}{\left(6-3\right)!3!}=\frac{6\text{*}5\text{*}4\text{*}3\text{*}2\text{*}1}{3!3!}=20$