Answer :
To determine which of the given options demonstrates how the number 70 is calculated using the combination pattern, we can recognize that 70 appears in the combination pattern as [tex]\(\binom{8}{4}\)[/tex].
The combination formula, also known as "n choose k," is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{(n - k)! k!} \][/tex]
Let's consider each option:
Option A: [tex]\(\binom{8}{4} = \frac{8!}{(8-4)! \cdot 4!}\)[/tex]
Here:
- [tex]\(n = 8\)[/tex]
- [tex]\(k = 4\)[/tex]
The formula simplifies to:
[tex]\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} \][/tex]
Calculating the factorial values, we get:
[tex]\[ 8! = 40320 \][/tex]
[tex]\[ 4! = 24 \][/tex]
Hence,
[tex]\[ \binom{8}{4} = \frac{40320}{24 \times 24} = \frac{40320}{576} = 70 \][/tex]
This confirms that Option A is correct.
Option B: [tex]\(\binom{70}{8} = \frac{70!}{(70-8)!8!}\)[/tex]
Here:
- [tex]\(n = 70\)[/tex]
- [tex]\(k = 8\)[/tex]
However, calculating [tex]\(\binom{70}{8}\)[/tex] does not fit the pattern of the given combination that results in 70.
Option C: [tex]\(\binom{8}{5} = \frac{8!}{(8-5)!5!}\)[/tex]
Here:
- [tex]\(n = 8\)[/tex]
- [tex]\(k = 5\)[/tex]
This would require us to compute:
[tex]\[ \binom{8}{5} = \frac{8!}{3! \cdot 5!} \][/tex]
This does not simplify to 70.
Option D: [tex]\(\binom{70}{4} = \frac{70!}{(70-4)!4!}\)[/tex]
Here:
- [tex]\(n = 70\)[/tex]
- [tex]\(k = 4\)[/tex]
Calculating [tex]\(\binom{70}{4}\)[/tex] would yield an extremely large number, not 70.
Thus, after considering all the options, the correct method to calculate 70 using the combination pattern is shown in Option A:
[tex]\[ \binom{8}{4} = \frac{8!}{(8-4)!4!} = 70 \][/tex]
The combination formula, also known as "n choose k," is given by:
[tex]\[ \binom{n}{k} = \frac{n!}{(n - k)! k!} \][/tex]
Let's consider each option:
Option A: [tex]\(\binom{8}{4} = \frac{8!}{(8-4)! \cdot 4!}\)[/tex]
Here:
- [tex]\(n = 8\)[/tex]
- [tex]\(k = 4\)[/tex]
The formula simplifies to:
[tex]\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} \][/tex]
Calculating the factorial values, we get:
[tex]\[ 8! = 40320 \][/tex]
[tex]\[ 4! = 24 \][/tex]
Hence,
[tex]\[ \binom{8}{4} = \frac{40320}{24 \times 24} = \frac{40320}{576} = 70 \][/tex]
This confirms that Option A is correct.
Option B: [tex]\(\binom{70}{8} = \frac{70!}{(70-8)!8!}\)[/tex]
Here:
- [tex]\(n = 70\)[/tex]
- [tex]\(k = 8\)[/tex]
However, calculating [tex]\(\binom{70}{8}\)[/tex] does not fit the pattern of the given combination that results in 70.
Option C: [tex]\(\binom{8}{5} = \frac{8!}{(8-5)!5!}\)[/tex]
Here:
- [tex]\(n = 8\)[/tex]
- [tex]\(k = 5\)[/tex]
This would require us to compute:
[tex]\[ \binom{8}{5} = \frac{8!}{3! \cdot 5!} \][/tex]
This does not simplify to 70.
Option D: [tex]\(\binom{70}{4} = \frac{70!}{(70-4)!4!}\)[/tex]
Here:
- [tex]\(n = 70\)[/tex]
- [tex]\(k = 4\)[/tex]
Calculating [tex]\(\binom{70}{4}\)[/tex] would yield an extremely large number, not 70.
Thus, after considering all the options, the correct method to calculate 70 using the combination pattern is shown in Option A:
[tex]\[ \binom{8}{4} = \frac{8!}{(8-4)!4!} = 70 \][/tex]
