Suppose we want to choose 2 letters, without replacement, from the 5 letters A, B, C, D, and E.
(a) How many ways can this be done, if the order of the choices is taken into consideration?
(b) How many ways can this be done, if the order of the choices is not taken into consideration?

Answer

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Answer

Calculate combinations: \(C(5,2) = \frac{5!}{2!3!} = \frac{5 \times 4}{2} = \boxed{10}\)

Steps

Step 1 ：\(a\) Calculate number of permutations: \(P(n,r) = \frac{n!}{(n-r)!}\)

Step 2 ：Substitute values: \(P(5,2) = \frac{5!}{(5-2)!}\)

Step 3 ：Calculate permutations: \(P(5,2) = \frac{5!}{3!} = 5 \times 4 = \boxed{20}\)

Step 4 ：\(b\) Calculate number of combinations: \(C(n,r) = \frac{n!}{r!(n-r)!}\)

Step 5 ：Substitute values: \(C(5,2) = \frac{5!}{2!(5-2)!}\)

Step 6 ：Calculate combinations: \(C(5,2) = \frac{5!}{2!3!} = \frac{5 \times 4}{2} = \boxed{10}\)