In order to find out the number of ways 7 letters can be selected from 12 letters, when order is not important, we can use the concept of combinations from combinatorial mathematics.
The formula for combinations is given as follows:
C(n, r) = n! / [(n - r)! * r!]
Here,
- n (total_letters) is the total number of elements from which elements will be chosen;
- r (letters_chosen) is the number of elements that are to be chosen;
- C(n, r) represents the number of combinations;
- "!" denotes the factorial of a number.
Let's calculate the combination:
First, let's plug the numbers into the formula:
C(12, 7) = 12! / [(12 - 7)! * 7!]
The factorial of a number is the product of that number and all the numbers below it down to 1. So here, we calculate the factorials:
- 12! equals 479,001,600
- 7! equals 5,040
- (12 - 7)! equals 5!
So we substitute these values into the formula:
C(12, 7) = 479,001,600 / [(5!) * 5,040]
5! equals 120.
So, C(12,7) = 479,001,600 / [120 * 5,040]
Finally, divide the numerator by the denominator gives us the number of ways 7 letters can be selected from 12 different letters.
So, there are 792 ways 7 letters can be chosen from the 12 distinct letters, when order does not matter.