Answer:
55
Explanation:
We want to know the coefficient of the x² in the expansion of (1+x)¹¹.
I learned about Pascal's triangle in Precalculus. This triangle will help you to see the coefficients of terms in larger binomial expansions.
See the attached jpg for Pascal's Triangle. The first row represents the 0th power. (1 + x)⁰ can be rewritten as (x+1)⁰. This is equal to 1 because anything raised to the 'zero' power is 1. Look at the second row. This represents (x+1)¹ = x + 1. Notice the coefficient of x is 1 and coefficient of the second number is also 1. There is no 'x' on the constant but the 1 is the coefficient of x⁰. The pattern with this triangle is that the number directly below two numbers is equal to their sum. This sum represents the coefficient of its term. Move from left to right on a single row, decreasing the power of x by 1 every jump.
The simplified form of (x+1)⁴ is x⁴ + 5x³ + 10x² + 5x + 1. Notice this is the 5th row of the triangle, the one that represents (a+b)⁴. Remember that the first row is for x⁰, so subtract 1 when using the triangle as a shortcut.
So, go down to the 12th row that represents the 11th power. We want the coefficient of the x² term. This is represented by the third and third to last numbers. So the coefficient of the x² term from the binomial expansion of (1+x)¹¹ is 55.
**This triangle is for the coefficients of binomial expansions.
Look up "Pascal's Triangle" for information on how to use this shortcut for the expansion of any binomial with coefficients for a.**