Chapter 5.8, Problem 55SE

a.

To determine

Find the combination $C{}_r$ for the given values of n and r.

Answer to Problem 55SE

The permutation $C{}_r$ for the given values of n and r is 56.

Explanation of Solution

Calculation:

The given information is that the value of n is 8 and the value of r is 3.

Combination:

The combination is the collection of r items selected at random without replacement from n items. It can be denoted as $C{}_r$.

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

Substitute 8 for n and 3 for r,

$\begin{array}{c}C{}_3=\frac{8!}{\left(8-3\right)!3!}\\ =\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(5\times 4\times 3\times 2\times 1\right)\left(3\times 2\times 1\right)}\\ =\frac{40,320}{720}\\ =56\end{array}$

Thus, the permutation $C{}_r$ for the given values of n and r is 56.

b.

To determine

Find the combination $C{}_r$ for the given values of n and r.

Answer to Problem 55SE

The permutation $C{}_r$ for the given values of n and r is 56.

Explanation of Solution

Calculation:

The given information is that the value of n is 8 and the value of r is 5.

Substitute 8 for n and 5 for r in $C{}_r$ formula,

$\begin{array}{c}C{}_5=\frac{8!}{\left(8-5\right)!5!}\\ =\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\left(5\times 4\times 3\times 2\times 1\right)}\\ =\frac{40,320}{720}\\ =56\end{array}$

Thus, the permutation $C{}_r$ for the given values of n and r is 56.

c.

To determine

Find the combination $C{}_r$ for the given values of n and r.

Answer to Problem 55SE

The permutation $C{}_r$ for the given values of n and r is 8.

Explanation of Solution

Calculation:

The given information is that the value of n is 8 and the value of r is 1.

Substitute 8 for n and 1 for r in $C{}_r$ formula,

$\begin{array}{c}C{}_7=\frac{8!}{\left(8-1\right)!1!}\\ =\frac{8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{7\times 6\times 5\times 4\times 3\times 2\times 1}\\ =\frac{40,320}{5,040}\\ =8\end{array}$

Thus, the permutation $C{}_r$ for the given values of n and r is 8.

d.

To determine

Find the combination $C{}_r$ for the given values of n and r.

Answer to Problem 55SE

The permutation $C{}_r$ for the given values of n and r is 1.

Explanation of Solution

Calculation:

The given information is that the value of n is 8 and the value of r is 8.

Substitute 8 for n and 8 for r in $C{}_r$ formula,

$\begin{array}{c}C{}_8=\frac{8!}{\left(8-8\right)!8!}\\ =\frac{8!}{0!8!}\\ =\frac{40,320}{\left(1\right)\left(40,320\right)}\\ =1\end{array}$

Thus, the permutation $C{}_r$ for the given values of n and r is 1.