Question

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

How many ways can this be done, if the order of the choices is taken into consideration?

Answer 1

To choose 7 letters from 12 distinct letters where order is considered, one must calculate the number of permutations using the formula P(12, 7) which equals 12! / 5!.

Step-by-step explanation:

Choosing 7 letters from 12 distinct letters where order matters is a permutation problem. Since the selection is without replacement, we use the permutation formula:

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

where n is the total number of letters and k is the number of letters chosen, resulting in 12! / (12-7)! permutations. This simplifies to:

P(12, 7) = 12! / 5! = 12 \u00d7 11 \u00d7 10 \u00d7 9 \u00d7 8 \u00d7 7 \u00d7 6

Calculating this gives us the total number of ways the 7 letters can be chosen from the 12 distinct letters considering the ordering.

Using factorials enables us to quantify the permutations systematically, similarly to how we can arrange four words into a valid sentence without exploring all combinations due to our intuition of grammar rules. In combinatorics, rules like factorials help in counting the possibilities without listing them.