Question

Suppose we want to choose 7 letters, without replacement, from 12 distinct letters.

(a) If the order of the choices is not taken into consideration, how many ways can this be done?
(b) If the order of the choices is taken into consideration, how many ways can this be done?

Answer 1

There are 792 ways to choose 7 letters without considering the order.

There are 3991680 ways to choose 7 letters while considering the order.

Choosing 7 letters from 12 distinct letters

a) Without considering order (combinations)

We have 12 distinct letters and we want to choose 7 of them without considering the order. This is a combination problem. We use the combination formula:

C(n, k) = n! / (k! * (n-k)!)

where:

n = total number of letters (12)

k = number of letters to choose (7)

Therefore, the number of ways to choose 7 letters without considering order is:

C(12, 7) = 12! / (7! * (12-7)!)

C(12, 7) = 792

b) Considering order (permutations):

If the order of the chosen letters matters, then this is a permutation problem. We use the permutation formula:

P(n, k) = n! / (n-k)!

where:

n = total number of letters (12)

k = number of letters to choose (7)

Therefore, the number of ways to choose 7 letters considering order is:

P(12, 7) = 12! / (12-7)!

P(12, 7) = 3991680