Evaluate the expression.
-$\frac { 9 ^ { C _ { 3 } } } { { } _ { 9 } P _ { 3 } }$
A) $\frac { 1 } { 3 }$
B) 6
C) 3
D) $\frac { 1 } { 6 }$

Question

Quizplus · Accepted Answer

The expression simplifies to $\frac{9C3}{9P3}$. $9C3$ calculates the number of combinations of 9 items taken 3 at a time, while $9P3$ calculates the number of permutations of 9 items taken 3 at a time. The formula for combinations is $nCk = \frac{n!}{k!(n-k)!}$ and for permutations is $nPk = \frac{n!}{(n-k)!}$. Plugging in the values, we get $9C3 = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!}$ and $9P3 = \frac{9!}{(9-3)!} = \frac{9!}{6!}$. Simplifying the expression $\frac{9C3}{9P3}$ gives $\frac{\frac{9!}{3!6!}}{\frac{9!}{6!}} = \frac{1}{3!} = \frac{1}{6}$, since $3! = 6$.

Evaluate the Expression 9C39P3\frac { 9 ^ { C _ { 3 } } } { { } _ { 9 } P _ { 3 } }9​P3​9C3​​

Evaluate the Expression $\frac { 9 ^ { C _ { 3 } } } { { } _ { 9 } P _ { 3 } }$