Question

If the order of objects is not of importance, how many ways can sixteen objects be selected two at a time? Why is this result different from the result when order matters?

Answer 1

To determine the number of ways to select two objects from sixteen without regard to order, we use the combinations formula C(16, 2), which results in 120 ways. This differs from permutations where order matters, which would result in more possible outcomes.

Step-by-step explanation:

When selecting objects and the order of selection does not matter, what we are looking for is the number of combinations. In the case of choosing two objects from a set of sixteen, we can use the combination formula, which is C(n, k) = n! / (k!(n - k)!), where 'n' is the total number of objects and 'k' is the number of objects to choose.

Applying the formula for our case: C(16, 2) = 16! / (2!(16 - 2)!) = 16! / (2!14!) = (16×15) / (2×1) = 120. There are 120 ways to select two objects from sixteen without regard to order.

This result is different from when order matters, which is a permutation, because in permutations every different order of selection counts as a distinct outcome. The formula for permutations is P(n, k) = n! / (n - k)!. However, for our combination scenario, each set of two objects is unique, no matter how they are ordered, thus resulting in fewer combinations.