Practice Binomial Theorem in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

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.

Each term of (a+b)n(a+b)^n picks 'aa' or 'bb' from each factor. (nk)\binom{n}{k} counts how many ways to pick kk bb's.

Showing a random 20 of 50 problems.

Example 1

medium
Expand (x+2)3(x + 2)^3 using the Binomial Theorem.

Example 2

easy
Write the coefficients in the expansion of (a+b)4(a+b)^4.

Example 3

medium
Sum of coefficients of (3x−2)5(3x-2)^5 when x=1x=1?

Example 4

medium
Use the binomial theorem to compute 11311^3 by writing 11=10+111=10+1.

Example 5

challenge
Prove that the sum of the coefficients in (a+b)n(a+b)^n equals 2n2^n.

Example 6

hard
Find the coefficient of x6x^6 in (1+x+x2)4(1+x+x^2)^4.

Example 7

challenge
Prove the Vandermonde identity: (m+nr)=∑k=0r(mk)(nr−k)\binom{m+n}{r}=\sum_{k=0}^r \binom{m}{k}\binom{n}{r-k}.

Example 8

medium
Find the coefficient of x4x^4 in (2x2+3)4(2x^2+3)^4.

Example 9

medium
What is ∑k=0n(nk)=?\sum_{k=0}^n \binom{n}{k}=?

Example 10

hard
Find the coefficient of x3x^3 in the expansion of (2x+3)5(2x + 3)^5.

Example 11

hard
In (2x−3y)4(2x-3y)^4, find the coefficient of x2y2x^2 y^2.

Example 12

challenge
Find the term containing x4x^4 in (x2+2x)5\left(x^2+\frac{2}{x}\right)^5.

Example 13

medium
Find the coefficient of x2x^2 in (2x+3)4(2x+3)^4.

Example 14

easy
Compute (52)\binom{5}{2}.

Example 15

medium
Find the coefficient of x2y3x^2y^3 in (x+y)5(x+y)^5.

Example 16

easy
Compute (73)\binom{7}{3}.

Example 17

easy
Compute (63)\binom{6}{3}.

Example 18

easy
Compute the binomial coefficient (42)\binom{4}{2}.

Example 19

easy
What is the coefficient of a2ba^2b in the expansion of (a+b)3(a+b)^3?

Example 20

medium
Expand (1+x)4(1+x)^4 and use it to evaluate (1.01)4(1.01)^4 to 4 decimal places.
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