Use the formula for nCr to solve Exercises 49–56. An election ballot asks voters to select three city commissioners from a group of six candidates. In how many ways can this be done?

Verified step by step guidance

1. Identify the problem as a combination problem because the order of selection does not matter when choosing city commissioners.
2. Recall the formula for combinations, which is $\mathrm{nCr}=\frac{n!}{r!\cdot (n-r)!}$, where $n$ is the total number of candidates and $r$ is the number of selections to be made.
3. Substitute the given values into the formula: $n=\; 6$ (candidates) and $r=\; 3$ (positions to fill).
4. Write the expression for the number of ways to choose 3 commissioners from 6 candidates as $\backslash binom\{6\}\{3\}\; =\; \backslash frac\{6!\}\{3!\; (6-3)!\}$.
5. Simplify the factorial expressions step-by-step to find the number of combinations, remembering that $n!\; =\; n\; \backslash times\; (n-1)\; \backslash times\; \backslash cdots\; \backslash times\; 1$.

Key Concepts

Combination Formula (nCr)

The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for counting selections where order does not matter.

Factorials

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 4! = 4 × 3 × 2 × 1 = 24. Factorials are used in permutations and combinations to calculate the total number of arrangements or selections.

Difference Between Combinations and Permutations

Combinations count selections where order does not matter, while permutations count arrangements where order matters. In this problem, since the order of selecting city commissioners is irrelevant, combinations are used to find the number of ways to choose the candidates.