Binomial Coefficient Math Example 3

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Example 3

medium
Calculate (83)\binom{8}{3} and explain what it counts.

Solution

  1. 1
    Apply the formula: (83)=8!3!(8−3)!=8!3!⋅5!\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8!}{3! \cdot 5!}.
  2. 2
    Simplify by canceling 5!5!: 8×7×63×2×1=3366=56\frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56.
  3. 3
    Interpretation: there are 56 ways to choose 3 items from a set of 8 when order does not matter. For example, 56 different 3-person committees from a group of 8 people.

Answer

(83)=56\binom{8}{3} = 56
The binomial coefficient (nk)\binom{n}{k} counts combinations — selections where order doesn't matter. We divide n!n! by k!k! (to remove the ordering of selected items) and (n−k)!(n-k)! (to remove the ordering of unselected items). Note that (83)=(85)=56\binom{8}{3} = \binom{8}{5} = 56 by symmetry.

About Binomial Coefficient

The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn distinct items without regard to order. It equals n!k!(n−k)!\frac{n!}{k!(n-k)!}.

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More Binomial Coefficient Examples

Example 1 medium

Calculate [formula] using the formula [formula], and verify by listing all combinations of 2 items f

Example 2 hard

A fair coin is flipped 5 times. Using [formula], find [formula] (exactly 3 heads).

Example 4 easy

Calculate: (a) [formula], (b) [formula], (c) [formula].

Example 5 hard

A committee of 3 is chosen from 8 people. How many possible committees exist? If one specific pair (

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