Answer :
To determine how many ways you can select 12 paintings out of a collection of 20, we use the concept of combinations. This is because the order in which we select the paintings does not matter.
The formula for combinations is:
[tex]\[ C(n, r) = \frac{n!}{r! \times (n-r)!} \][/tex]
Where:
- [tex]\( n \)[/tex] = total number of items to choose from (in this case, 20 paintings).
- [tex]\( r \)[/tex] = number of items to choose (in this case, 12 paintings).
Step-by-step calculation:
1. Identify [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
- [tex]\( n = 20 \)[/tex]
- [tex]\( r = 12 \)[/tex]
2. Substitute the values into the combinations formula:
[tex]\[
C(20, 12) = \frac{20!}{12! \times (20-12)!}
\][/tex]
3. Calculate the factorials:
- [tex]\( 20! \)[/tex] is the product of all positive integers up to 20.
- [tex]\( 12! \)[/tex] is the product of all positive integers up to 12.
- [tex]\( (20-12)! = 8! \)[/tex] is the product of all positive integers up to 8.
4. Simplify the expression:
[tex]\[
C(20, 12) = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\][/tex]
5. Calculate the final result:
After simplifying the expression by canceling common terms in the numerator and the denominator, the result is 125970.
Therefore, there are 125970 different ways to select 12 paintings from a collection of 20.
The formula for combinations is:
[tex]\[ C(n, r) = \frac{n!}{r! \times (n-r)!} \][/tex]
Where:
- [tex]\( n \)[/tex] = total number of items to choose from (in this case, 20 paintings).
- [tex]\( r \)[/tex] = number of items to choose (in this case, 12 paintings).
Step-by-step calculation:
1. Identify [tex]\( n \)[/tex] and [tex]\( r \)[/tex]:
- [tex]\( n = 20 \)[/tex]
- [tex]\( r = 12 \)[/tex]
2. Substitute the values into the combinations formula:
[tex]\[
C(20, 12) = \frac{20!}{12! \times (20-12)!}
\][/tex]
3. Calculate the factorials:
- [tex]\( 20! \)[/tex] is the product of all positive integers up to 20.
- [tex]\( 12! \)[/tex] is the product of all positive integers up to 12.
- [tex]\( (20-12)! = 8! \)[/tex] is the product of all positive integers up to 8.
4. Simplify the expression:
[tex]\[
C(20, 12) = \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}
\][/tex]
5. Calculate the final result:
After simplifying the expression by canceling common terms in the numerator and the denominator, the result is 125970.
Therefore, there are 125970 different ways to select 12 paintings from a collection of 20.
