Question

which number do you expect to be greater, a combination or a permutation of n objects taken r at a time? why? choose the correct answer below. a. there are usually more permutations because like objects are indistinguishable in combinations. b. there are usually more permutations because one or more permutations can be made from each combination. c. there are usually more combinations because like objects are indistinguishable in permutations. d. there are usually more combinations because one or more combinations can be made from each permutation.

Explanation:

To determine whether permutations or combinations of \( n \) objects taken \( r \) at a time are greater, we recall the formulas:

• Permutation: \( P(n, r) = \frac{n!}{(n - r)!} \)
• Combination: \( C(n, r) = \frac{n!}{r!(n - r)!} \)

Dividing \( P(n, r) \) by \( C(n, r) \) gives \( \frac{P(n, r)}{C(n, r)} = r! \), which is always \( \geq 2 \) for \( r \geq 2 \) (and \( > 1 \) for \( r \geq 1 \), since \( r! \geq 1 \), but for \( r \geq 2 \), \( r! \geq 2 \)). This means \( P(n, r) = r! \cdot C(n, r) \), so permutations are greater because each combination can be rearranged into \( r! \) permutations (i.e., one or more permutations come from each combination). Option B correctly states this logic.

Answer:

B. There are usually more permutations because one or more permutations can be made from each combination.