Binomial Theorem Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium
What is (62)\binom{6}{2}?

Solution

  1. 1
    (62)=6!2!⋅4!=6×52×1=15\binom{6}{2} = \frac{6!}{2! \cdot 4!} = \frac{6 \times 5}{2 \times 1} = 15.
  2. 2
    This means the x4x^4 term in (a+b)6(a+b)^6 has coefficient 15.

Answer

1515
The binomial coefficient (nk)\binom{n}{k} counts the number of ways to choose kk items from nn. It also gives the coefficient of the (k+1)(k+1)-th term in the binomial expansion.

About Binomial Theorem

The binomial theorem gives the expansion of (a + b)^n as a sum of terms involving binomial coefficients: (a+b)^n = sum of C(n,k) * a^(n-k) * b^k. Each coefficient (nk)\binom{n}{k} counts the number of ways to choose kk copies of bb from nn factors.

Learn more about Binomial Theorem →

More Binomial Theorem Examples

Example 1 medium

Expand [formula] using the Binomial Theorem.

Example 2 hard

Find the coefficient of [formula] in the expansion of [formula].

Example 3 easy

Expand [formula] using Pascal's triangle.

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