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Answer :
To solve for [tex]\( C(9,7) \)[/tex], we use the combination formula, which is:
[tex]\[
C(n, k) = \frac{n!}{k! \times (n-k)!}
\][/tex]
where [tex]\( n \)[/tex] is the total number of items to choose from, and [tex]\( k \)[/tex] is the number of items to choose. Given [tex]\( n = 9 \)[/tex] and [tex]\( k = 7 \)[/tex], we can plug these values into the formula:
1. Calculate the factorial of 9:
[tex]\[
9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
2. Calculate the factorial of 7:
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
3. Calculate the factorial of (9 - 7):
[tex]\[
(9 - 7)! = 2! = 2 \times 1
\][/tex]
4. Substitute these values into the combination formula:
[tex]\[
C(9, 7) = \frac{9!}{7! \times 2!}
\][/tex]
5. Simplify this expression:
- Evaluate [tex]\( 9! \)[/tex] and [tex]\( 7! \)[/tex]:
[tex]\[
\frac{9 \times 8 \times 7!}{7! \times 2!} = \frac{9 \times 8}{2 \times 1}
\][/tex]
- Calculate the division:
[tex]\[
\frac{72}{2} = 36
\][/tex]
So, [tex]\( C(9,7) = 36 \)[/tex].
[tex]\[
C(n, k) = \frac{n!}{k! \times (n-k)!}
\][/tex]
where [tex]\( n \)[/tex] is the total number of items to choose from, and [tex]\( k \)[/tex] is the number of items to choose. Given [tex]\( n = 9 \)[/tex] and [tex]\( k = 7 \)[/tex], we can plug these values into the formula:
1. Calculate the factorial of 9:
[tex]\[
9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
2. Calculate the factorial of 7:
[tex]\[
7! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
3. Calculate the factorial of (9 - 7):
[tex]\[
(9 - 7)! = 2! = 2 \times 1
\][/tex]
4. Substitute these values into the combination formula:
[tex]\[
C(9, 7) = \frac{9!}{7! \times 2!}
\][/tex]
5. Simplify this expression:
- Evaluate [tex]\( 9! \)[/tex] and [tex]\( 7! \)[/tex]:
[tex]\[
\frac{9 \times 8 \times 7!}{7! \times 2!} = \frac{9 \times 8}{2 \times 1}
\][/tex]
- Calculate the division:
[tex]\[
\frac{72}{2} = 36
\][/tex]
So, [tex]\( C(9,7) = 36 \)[/tex].
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