Chapter 11.7, Problem 3C

To determine

To calculate: The number of combinations.

Answer to Problem 3C

  ${}_8{C}_5\text{ is equal to 56}\text{.}$

Explanation of Solution

Given information:

Consider the information provided in the question:

  ${}_8{C}_5$

Formula used:

To find the number of combinations of $n$ objects taken $r$ at a time, divide the number of permutations of $n$ objects taken $r$ at a time by $r!$

  ${}_n{C}_r=\frac{{}_n{P}_r}{r!}$

Calculation:

As per the problem,

  ${}_8{C}_5$

To calculate the number of combinations, use formula

  ${}_n{C}_r=\frac{{}_n{P}_r}{r!}$

Substitute $\text{8 for }n\text{ and 5 for }r$

  ${}_8{C}_5=\frac{{}_8{P}_5}{5!}$

  $=\frac{8\cdot 7\cdot 6\cdot 5\cdot 4}{5\cdot 4\cdot 3\cdot 2\cdot 1}$ [Write ${}_8{P}_5$ and $5!$ as products]

  $=56$ [Divide out common factors and simplify]

Hence,

  ${}_8{C}_5\text{ is equal to 56}\text{.}$