Answer :
To find the number of ways to select 12 paintings out of a collection of 20, you can use the combination formula, which determines how many ways you can choose a group of items from a larger set without regard to the order of selection. The formula for combinations is:
[tex]\[
{}_nC_r = \frac{n!}{r!(n-r)!}
\][/tex]
In this case, [tex]\( n \)[/tex] is the total number of paintings, which is 20, and [tex]\( r \)[/tex] is the number of paintings you want to select, which is 12. Plugging these values into the formula gives:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12!(20-12)!}
\][/tex]
This simplifies to:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12! \times 8!}
\][/tex]
Now, compute the factorials and substitute them into the equation:
- [tex]\( 20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12! \)[/tex]
- [tex]\( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)[/tex]
- [tex]\( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)[/tex]
Cancel out the [tex]\( 12! \)[/tex] in the numerator and the denominator:
[tex]\[
{}_{20}C_{12} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\][/tex]
After performing the above calculations, you will find that:
[tex]\[
{}_{20}C_{12} = 125970
\][/tex]
Thus, there are 125,970 ways to choose 12 paintings out of 20.
[tex]\[
{}_nC_r = \frac{n!}{r!(n-r)!}
\][/tex]
In this case, [tex]\( n \)[/tex] is the total number of paintings, which is 20, and [tex]\( r \)[/tex] is the number of paintings you want to select, which is 12. Plugging these values into the formula gives:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12!(20-12)!}
\][/tex]
This simplifies to:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12! \times 8!}
\][/tex]
Now, compute the factorials and substitute them into the equation:
- [tex]\( 20! = 20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12! \)[/tex]
- [tex]\( 12! = 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)[/tex]
- [tex]\( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)[/tex]
Cancel out the [tex]\( 12! \)[/tex] in the numerator and the denominator:
[tex]\[
{}_{20}C_{12} = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\][/tex]
After performing the above calculations, you will find that:
[tex]\[
{}_{20}C_{12} = 125970
\][/tex]
Thus, there are 125,970 ways to choose 12 paintings out of 20.
