Which of the following demonstrates how 70 is calculated using the combination pattern?

[tex]\[
\begin{array}{lllllllll}
1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1
\end{array}
\][/tex]

A. [tex]\({ }_8 C_4=\binom{8}{4}=\frac{8!}{(8-4)!4!}=70\)[/tex]
B. [tex]\({ }_{70} C_8=\binom{70}{8}=\frac{70!}{(70-8)!8!}=70\)[/tex]
C. [tex]\({ }_8 C_5=\binom{8}{5}=\frac{8!}{(8-5)!5!}=70\)[/tex]
D. [tex]\({ }_{70} C_4=\binom{70}{4}=\frac{70!}{(70-4)!4!}=70\)[/tex]



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]