Evaluate the expression. $$ { }_{7} \mathrm{P}_{4} $$ The solution is $$ \square $$.

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UpStudy AI · Accepted Answer

1. Use the permutation formula:
   $$
   {}_nP_k = \frac{n!}{(n-k)!}
   $$
2. Substitute $$ n = 7 $$ and $$ k = 4 $$:
   $$
   {}_7P_4 = \frac{7!}{(7-4)!} = \frac{7!}{3!}
   $$
3. Write $$ 7! $$ as:
   $$
   7! = 7 \times 6 \times 5 \times 4 \times 3!
   $$
4. Substitute into the expression:
   $$
   {}_7P_4 = \frac{7 \times 6 \times 5 \times 4 \times 3!}{3!}
   $$
5. Cancel $$ 3! $$ in the numerator and denominator:
   $$
   {}_7P_4 = 7 \times 6 \times 5 \times 4
   $$
6. Compute the product:
   $$
   7 \times 6 = 42, \quad 42 \times 5 = 210, \quad 210 \times 4 = 840
   $$

The solution is $$ \boxed{840} $$.
The solution is 840.

Evaluate the expression. \( { }_{7} \mathrm{P}_{4} \) The solution is \( \square \).

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