Sure, let’s solve the permutation problem step by step:
Given the problem:
[tex]\[
\frac{{ }_6 P_3}{{ }_{10} P_4}
\][/tex]
We need to calculate the permutation values first.
### Step 1: Calculate the permutation [tex]\( {}_6 P_3 \)[/tex]
Permutation [tex]\( {}_n P_r \)[/tex] is calculated using the formula:
[tex]\[
{}_n P_r = \frac{n!}{(n-r)!}
\][/tex]
For [tex]\( {}_6 P_3 \)[/tex]:
[tex]\[
{}_6 P_3 = \frac{6!}{(6-3)!} = \frac{6!}{3!}
\][/tex]
Calculate the factorials:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
Now, compute the permutation:
[tex]\[
{}_6 P_3 = \frac{720}{6} = 120
\][/tex]
### Step 2: Calculate the permutation [tex]\( {}_{10} P_4 \)[/tex]
Using the same formula for [tex]\( {}_{10} P_4 \)[/tex]:
[tex]\[
{}_{10} P_4 = \frac{10!}{(10-4)!} = \frac{10!}{6!}
\][/tex]
Calculate the factorials:
[tex]\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\][/tex]
[tex]\[
6! (as calculated before) = 720
\][/tex]
Now, compute the permutation:
[tex]\[
{}_{10} P_4 = \frac{3,628,800}{720} = 5040
\][/tex]
### Step 3: Perform the indicated calculation
Now we have:
[tex]\[
{}_6 P_3 = 120
\][/tex]
[tex]\[
{}_{10} P_4 = 5040
\][/tex]
Thus, the calculation is:
[tex]\[
\frac{{ }_6 P_3}{{ }_{10} P_4} = \frac{120}{5040}
\][/tex]
Simplify the fraction:
[tex]\[
\frac{120}{5040} = 0.023809523809523808
\][/tex]
### Final Answer:
[tex]\[ \frac{{ }_6 P_3}{{ }_{10} P_4} = 0.023809523809523808 \][/tex]
So, the result is approximately [tex]\( 0.02381 \)[/tex].
