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Answer :
Certainly! To determine how many ways you can select 12 paintings out of a collection of 20, you can use the concept of combinations. Combinations are used when the order of selection does not matter.
The formula for combinations is:
[tex]\[
{}_nC_r = \frac{n!}{r!(n-r)!}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items to choose from (20 paintings).
- [tex]\( r \)[/tex] is the number of items to select (12 paintings).
- [tex]\( n! \)[/tex] (n factorial) is the product of all positive integers up to [tex]\( n \)[/tex].
Step-by-step solution:
1. Identify Values:
- Total number of paintings, [tex]\( n = 20 \)[/tex].
- Number of paintings to select, [tex]\( r = 12 \)[/tex].
2. Apply the Combination Formula:
- Substitute the values into the formula:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12! \times 8!}
\][/tex]
3. Calculate Factorials:
- [tex]\( 20! \)[/tex] is the product of all numbers from 1 to 20.
- [tex]\( 12! \)[/tex] is the product of all numbers from 1 to 12.
- [tex]\( 8! \)[/tex] is the product of all numbers from 1 to 8.
4. Simplify the Expression:
- Once you substitute the actual numbers from the factorial calculations above, the expression simplifies to a single number.
5. Final Calculation:
- By carrying out the division and simplifications step by step, you reach the final solution.
Thus, there are 125970 different ways to select 12 paintings from a group of 20 to hang in your art gallery.
The formula for combinations is:
[tex]\[
{}_nC_r = \frac{n!}{r!(n-r)!}
\][/tex]
Where:
- [tex]\( n \)[/tex] is the total number of items to choose from (20 paintings).
- [tex]\( r \)[/tex] is the number of items to select (12 paintings).
- [tex]\( n! \)[/tex] (n factorial) is the product of all positive integers up to [tex]\( n \)[/tex].
Step-by-step solution:
1. Identify Values:
- Total number of paintings, [tex]\( n = 20 \)[/tex].
- Number of paintings to select, [tex]\( r = 12 \)[/tex].
2. Apply the Combination Formula:
- Substitute the values into the formula:
[tex]\[
{}_{20}C_{12} = \frac{20!}{12!(20-12)!} = \frac{20!}{12! \times 8!}
\][/tex]
3. Calculate Factorials:
- [tex]\( 20! \)[/tex] is the product of all numbers from 1 to 20.
- [tex]\( 12! \)[/tex] is the product of all numbers from 1 to 12.
- [tex]\( 8! \)[/tex] is the product of all numbers from 1 to 8.
4. Simplify the Expression:
- Once you substitute the actual numbers from the factorial calculations above, the expression simplifies to a single number.
5. Final Calculation:
- By carrying out the division and simplifications step by step, you reach the final solution.
Thus, there are 125970 different ways to select 12 paintings from a group of 20 to hang in your art gallery.
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