Question

Suppose we want to choose 7 letters, without replacement, from 12 distinct letters. (a) How many ways can this be done, if the order of the choices is not relevant?

Answer 1

3991680 or 792

Explanation:

12P7 = 3991680

or

12C7 = 792

P = permutation

C = combination

Sorry, I am not sure which answer is correct or if any of the answers are correct.

Answer 2

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.