To solve the given expression [tex]\(\frac{{ }^5 C_3}{{ }^{20} C_5}\)[/tex], let's work through the steps for finding each combination and then perform the division.
Step 1: Understanding the Combination Formula
The formula for a combination, often denoted as [tex]\( nCr \)[/tex], is calculated using:
[tex]\[ nCr = \frac{n!}{r!(n-r)!} \][/tex]
where [tex]\( n! \)[/tex] represents the factorial of [tex]\( n \)[/tex], and [tex]\( r \)[/tex] is the number of elements to choose from [tex]\( n \)[/tex] items.
Step 2: Calculating [tex]\( { }^5C_3 \)[/tex]
Using the combination formula:
[tex]\[ { }^5C_3 = \frac{5!}{3!(5-3)!} = \frac{5!}{3! \cdot 2!} \][/tex]
Here, [tex]\( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)[/tex], [tex]\( 3! = 3 \times 2 \times 1 = 6 \)[/tex], and [tex]\( 2! = 2 \times 1 = 2 \)[/tex].
[tex]\[
{ }^5C_3 = \frac{120}{6 \cdot 2} = \frac{120}{12} = 10
\][/tex]
Therefore, [tex]\( { }^5C_3 = 10 \)[/tex].
Step 3: Calculating [tex]\( { }^{20}C_5 \)[/tex]
Using the combination formula:
[tex]\[ { }^{20}C_5 = \frac{20!}{5!(20-5)!} = \frac{20!}{5! \cdot 15!} \][/tex]
Calculating factorials directly can be tedious and large, but we know from the provided calculations that:
[tex]\[
{ }^{20}C_5 = 15504
\][/tex]
Step 4: Dividing the Two Combinations
Now that we have both combinations:
[tex]\[ { }^5C_3 = 10 \][/tex]
[tex]\[ { }^{20}C_5 = 15504 \][/tex]
We perform the division:
[tex]\[ \frac{{ }^5C_3}{{ }^{20}C_5} = \frac{10}{15504} \][/tex]
The result of this division is approximately:
[tex]\[
0.0006449948400412797
\][/tex]
Conclusion
Thus, the detailed solution for the given expression [tex]\(\frac{{ }^5 C_3}{{ }^{20} C_5}\)[/tex] results in:
[tex]\[
\frac{10}{15504} \approx 0.0006449948400412797
\][/tex]
