Questions: 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 matters? (b) How many ways can this be done, if the order of the choices does not matter?

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 matters?

(b) How many ways can this be done, if the order of the choices does not matter?

Transcript text: 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 matters? $\square$ (b) How many ways can this be done, if the order of the choices does not matter?

Solution

Solution Steps

Step 1: Identify the Problem Type

Given the problem to find the number of ways to choose $k=2$ items from $n=5$ distinct items, where order matters, we are dealing with a permutation problem.

Step 2: Apply the Permutation Formula

The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$.

Step 3: Perform the Calculation

Substituting the given values, we get $P(5, 2) = \frac{5!}{3!} = 20$.

Final Answer:

The number of ways to choose $k=2$ items from $n=5$ distinct items, where order matters, is 20.

Step 1: Identify the Problem Type

Given the problem to find the number of ways to choose $k=2$ items from $n=5$ distinct items, where order does not matter, we are dealing with a combination problem.

Step 2: Apply the Combination Formula

The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$, also known as the binomial coefficient.

Step 3: Perform the Calculation

Substituting the given values, we get $C(5, 2) = \frac{5!}{2!(3)!} = 10$.

Final Answer:

The number of ways to choose $k=2$ items from $n=5$ distinct items, where order does not matter, is 10.

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