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College Algebra

Binomial Theorem

Lesson

Expanding (a+b)n(a + b)^n by hand gets tedious fast. The binomial theoremgives you every term’s coefficient without multiplying everything out.

The formula

(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

(nk)\binom{n}{k}— read “n choose k” — is the coefficient. It also counts ways to choose kk things from nn:

(nk)=n!k!(n−k)!\binom{n}{k} = \frac{n!}{k!(n - k)!}

Pascal’s triangle

Row nn gives the coefficients of (a+b)n(a + b)^n. Each entry is the sum of the two above it.

Row 0:        1
Row 1:       1 1
Row 2:      1 2 1
Row 3:     1 3 3 1
Row 4:    1 4 6 4 1
Row 5:   1 5 10 10 5 1
Row 6:  1 6 15 20 15 6 1

Worked example 1

(x+1)4=x4+4x3+6x2+4x+1(x + 1)^4 = x^4 + 4x^3 + 6x^2 + 4x + 1

Row 4 of Pascal’s triangle: 1, 4, 6, 4, 1.

Worked example 2 — pick out one coefficient

Coefficient of x3x^3 in (2x+1)4(2x + 1)^4:

(43)⋅(2)3⋅(1)1=4⋅8⋅1=32\binom{4}{3} \cdot (2)^3 \cdot (1)^1 = 4 \cdot 8 \cdot 1 = 32

Watch the signs

For (a−b)n(a - b)^n, treat it as (a+(−b))n(a + (-b))^n. Odd powers of −b-b are negative; even powers are positive.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

(42)\binom{4}{2}

Problem 2

(52)\binom{5}{2}

Problem 3

(63)\binom{6}{3}

Problem 4

(31)\binom{3}{1}

Practice

Standard problems matching the lesson.

Problem 5

(72)\binom{7}{2}

Problem 6

(83)\binom{8}{3}

Problem 7

Coefficient of x3 in (x+1)5\text{Coefficient of } x^3 \text{ in } (x + 1)^5

Problem 8

Coefficient of x2 in (x+1)4\text{Coefficient of } x^2 \text{ in } (x + 1)^4

Problem 9

Coefficient of x3 in (x+2)4\text{Coefficient of } x^3 \text{ in } (x + 2)^4

Problem 10

Coefficient of x2 in (x+1)5\text{Coefficient of } x^2 \text{ in } (x + 1)^5

Problem 11

Coefficient of x in (1+x)7\text{Coefficient of } x \text{ in } (1 + x)^7

Problem 12

Coefficient of x4 in (x+1)6\text{Coefficient of } x^4 \text{ in } (x + 1)^6

Problem 13

Coefficient of x3 in (2x+1)4\text{Coefficient of } x^3 \text{ in } (2x + 1)^4

Problem 14

Number of terms in (x+y)7\text{Number of terms in } (x + y)^7

Problem 15

Sum of binomial coefficients in (1+1)4\text{Sum of binomial coefficients in } (1 + 1)^4

Problem 16

(100)\binom{10}{0}

Problem 17

Ways to choose 3 from 8?

Problem 18

5-bit strings with exactly 2 ones?

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

(94)\binom{9}{4}

Problem 20

Coefficient of x2 in (3x+1)4\text{Coefficient of } x^2 \text{ in } (3x + 1)^4

Problem 21

Coefficient of x3 in (x−1)5\text{Coefficient of } x^3 \text{ in } (x - 1)^5

Problem 22

Coefficient of x2 in (x−2)4\text{Coefficient of } x^2 \text{ in } (x - 2)^4

Problem 23

Coefficient of x in (2+x)6\text{Coefficient of } x \text{ in } (2 + x)^6

Problem 24

Constant term in (x+3)4\text{Constant term in } (x + 3)^4

Problem 25

(103)\binom{10}{3}

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