If the binomial (x+y) ^7 were expanded, what would be the coefficient of each term?

Term coefficient

x⁷ 1 x⁶y 7 x⁵y² 21 x⁴y³ 35 x³y⁴ 35 x²y⁵ 21 xy⁶ 7 y⁷ 1

Explanation:

You can use Pascal's triangle to predict the coefficient of each term in a binomial expansion.

Since the binomial has exponent 7, the expanded expression will have 8 terms: (x + y) ⁰ has 1 term, (x + y) ¹ has two terms, (x + y) ² has three terms, (x + y) ³ has four terms, and so on.

The Pascal triangle for 8 terms has 8 rows and they are:

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 row 7

1 7 21 35 35 21 7 1 row 8

So, the coefficients, in order, are the numbers from the row 8: 1, 7, 21, 35, 35, 21, 7, and 1.

And the terms in order are: x⁷y⁰, x⁶y¹, x⁵y², x⁴y³, x³y⁴, x²y⁵, x¹y⁶, and x⁰y⁷.

With that, you can write the coefficient of each term:

Term coefficient

x⁷y⁰ = x⁷ 1

x⁶y¹ = x⁶y 7

x⁵y² 21

x⁴y³ 35

x³y⁴ 35

x²y⁵ 21

x¹y⁶ = xy⁶ 7

x⁰y⁷ = y⁷ 1