Sure, let's walk through the steps to solve the given fraction [tex]\(\frac{{ }^5 C_3}{{ }^{20} C_5}\)[/tex].
1. Calculate [tex]\({ }^5 C_3\)[/tex]:
[tex]\[
{ }^5 C_3 = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!}
\][/tex]
Breaking down the factorials:
[tex]\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
[tex]\[
3! = 3 \times 2 \times 1 = 6
\][/tex]
[tex]\[
2! = 2 \times 1 = 2
\][/tex]
Substituting back, we get:
[tex]\[
{ }^5 C_3 = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10
\][/tex]
2. Calculate [tex]\({ }^{20} C_5\)[/tex]:
[tex]\[
{ }^{20} C_5 = \frac{20!}{5!(20-5)!} = \frac{20!}{5! \cdot 15!}
\][/tex]
Breaking down [tex]\(20!\)[/tex]:
[tex]\[
20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15!
\][/tex]
Substituting back:
[tex]\[
{ }^{20} C_5 = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15!}{5! \times 15!} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5!}
\][/tex]
[tex]\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\][/tex]
Substituting [tex]\(5!\)[/tex]:
[tex]\[
{ }^{20} C_5 = \frac{20 \times 19 \times 18 \times 17 \times 16}{120}
\][/tex]
Performing this calculation:
[tex]\[
\frac{20 \times 19 \times 18 \times 17 \times 16}{120} = 15504
\][/tex]
3. Calculate the fraction:
[tex]\[
\frac{{ }^5 C_3}{{ }^{20} C_5} = \frac{10}{15504}
\][/tex]
Performing the division gives:
[tex]\[
\frac{10}{15504} = 0.0006449948400412797
\][/tex]
So, the fraction [tex]\(\frac{{ }^5 C_3}{{ }^{20} C_5}\)[/tex] equals approximately 0.0006449948400412797.
