Binomial Coefficient Math Example 1

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Example 1

medium
Calculate (62)\binom{6}{2} using the formula (nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n-k)!}, and verify by listing all combinations of 2 items from {A,B,C,D,E,F}\{A, B, C, D, E, F\}.

Solution

  1. 1
    Apply formula: (62)=6!2!(6−2)!=6!2!⋅4!=6×5×4!2×1×4!=302=15\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5 \times 4!}{2 \times 1 \times 4!} = \frac{30}{2} = 15
  2. 2
    List all 2-item combinations from \{A,B,C,D,E,F\}: AB, AC, AD, AE, AF, BC, BD, BE, BF, CD, CE, CF, DE, DF, EF
  3. 3
    Count: 15 combinations ✓
  4. 4
    Confirms formula gives the correct count

Answer

(62)=15\binom{6}{2} = 15. Verified by listing all 15 two-item combinations.
The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn without regard to order. Order doesn't matter in combinations (unlike permutations). The formula cancels repeated arrangements via the k!k! in the denominator.

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 2 hard

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

Example 3 medium

Calculate [formula] and explain what it counts.

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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