## Binomial Coefficients and the Binomial Theorem

When a binomial is raised to whole number powers, the coefficients of the terms in the expansion form a pattern.

**These expressions exhibit many patterns:**

- Each expansion has one more term than the power on the binomial.
- The sum of the exponents in each term in the expansion is the same as the power on the binomial.
- The powers on
*a* in the expansion decrease by 1 with each successive term, while the powers on *b* increase by 1.
- The coefficients form a symmetrical pattern.
- Each coefficient entry below the second row is the sum of the closest pair of numbers in the line directly above it.

This triangular array is called **Pascal's triangle,** named after the French mathematician Blaise Pascal.

Pascal's triangle can be extended to find the coefficients for raising a binomial to any whole number exponent. This same array could be expressed using the factorial symbol, as shown in the following.

In general,

The symbol , called the **binomial coefficient,** is defined as follows:

Therefore,

This could be further condensed using sigma notation.

This formula is known as the **binomial theorem.**

Example 1

Use the binomial theorem to express ( *x* + *y*) ^{7} in expanded form.

Notice the following pattern:

In general, the *k*th term of any binomial expansion can be expressed as follows:

Example 2

Find the tenth term of the expansion ( *x* + *y*) ^{13}

Since *n* = 13 and *k* = 10,