Question

suppose we want to choose 2 objects, without replacement, from the 4 objects pencil, eraser, desk, and chair.

(if necessary, consult a list of formulas.)

(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?

Explanation:

Identify the parameters of the problem

We are choosing \(r = 2\) objects from a set of \(n = 4\) distinct objects: pencil, eraser, desk, and chair. The selection is done without replacement.

Calculate selections when order matters

Using the Probability of Events foundation, we find the number of ordered arrangements. When order is taken into consideration, we use permutations:

$$ P(n, r) = \frac{n!}{(n-r)!} $$

Substituting \(n = 4\) and \(r = 2\):

$$ P(4, 2) = \frac{4!}{(4-2)!} = \frac{24}{2} = 12 $$

Calculate selections when order does not matter

When order is not taken into consideration, we find the number of subsets. We use combinations:

$$ C(n, r) = \frac{n!}{r!(n-r)!} $$

Substituting \(n = 4\) and \(r = 2\):

$$ C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{24}{2 \times 2} = 6 $$

Answer:

Question (a)

12

Question (b)

6