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------------------------------------------------ In how many ways can you select 12 paintings out of a collection of 20 to hang in your art gallery?

[tex]{}_{n}C_{r} = \frac{n!}{r!(n-r)!}[/tex]

Answer :

To determine the number of ways to choose 12 paintings from a collection of 20, we use the combination formula:

[tex]$$
\binom{n}{r} = \frac{n!}{r!(n-r)!}
$$[/tex]

Here, [tex]$n = 20$[/tex] and [tex]$r = 12$[/tex], so the formula becomes:

[tex]$$
\binom{20}{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12! \times 8!}
$$[/tex]

The values of the factorials are:

- [tex]$20! = 2432902008176640000$[/tex]
- [tex]$12! = 479001600$[/tex]
- [tex]$8! = 40320$[/tex]

Substituting these into the formula:

[tex]$$
\binom{20}{12} = \frac{2432902008176640000}{479001600 \times 40320} = 125970
$$[/tex]

Thus, the number of ways to select 12 paintings out of 20 is:

[tex]$$
\boxed{125970}
$$[/tex]

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