Question 1 of 17 , Step 1 of 1
$0 / 17$
Correct
Evaluate the following expression, Express your answer as a fraction or a decimal number rounded to four decimal places.
\[
\frac{{ }_{13} P_{7}}{{ }_{13} C_{5}}
\]
Answer
How to enter your answer (opens in new window)

Step 1 ：The given expression is a ratio of a permutation to a combination. The permutation $_{13}P_{7}$ represents the number of ways to arrange 7 items out of 13, and the combination $_{13}C_{5}$ represents the number of ways to choose 5 items out of 13.

Step 2 ：The formula for permutation is $_nP_r = n! / (n-r)!$ and for combination is $_nC_r = n! / [(n-r)! * r!]$.

Step 3 ：So, we need to calculate the values of $_{13}P_{7}$ and $_{13}C_{5}$ and then divide the permutation by the combination.

Step 4 ：Calculate permutation: $_{13}P_{7} = 8648640.0$

Step 5 ：Calculate combination: $_{13}C_{5} = 1287.0$

Step 6 ：Calculate ratio: $\frac{{ }_{13} P_{7}}{{ }_{13} C_{5}} = 6720.0$

Step 7 ：Final Answer: \(\boxed{6720}\)