<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Study Guide. Go to the latest version.

# Binomial Theorem and Expansions

## Expansion of binomials raised to a power using combinations.

Estimated22 minsto complete
%
Progress
Practice Binomial Theorem and Expansions

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated22 minsto complete
%
Combinations and the Binomial Theorem

### Vocabulary

##### Complete the chart.
 Word Definition Combination ____________________________________________________________ Factorial ____________________________________________________________ Binomial Theorem ____________________________________________________________ _______________ the process of raising a binomial such as (x + 2) to a power _______________ a pyramid of sorts constructed with the coefficients of binomials as they are expanded

### Factorials and Combinations

Expand n!=\begin{align*}n! =\end{align*} ___________________________________

Simplify and Evaluate the Factorials:

1. (a+3)!(a+4)!\begin{align*}\frac{(a + 3)!}{(a + 4)!}\end{align*}
2. 5!4!2!\begin{align*}\frac{5!}{4! 2!}\end{align*}
3. 11!7!\begin{align*}\frac{11!}{7!}\end{align*}

Solve

1. 9C7\begin{align*} _{9}C_{7}\end{align*}
2. 5C2\begin{align*} _{5}C_{2}\end{align*}
3. The local TV station forecasts a 14% chance of rain every day for the next week. What is the probability that it will rain on exactly 4 out of the next 7 days?

.

#### Binomial Theorem and Expansions

Pascal's Triangle can help expand binomials.

It follows the pattern (nr1)+(nr)=(n+1r)\begin{align*}\binom{n} {r - 1} + \binom{n} {r} = \binom{n + 1} {r}\end{align*} .

.

When you expand ( x + y ) n , the exponents of x ______________ whille the exponents of y ______________.

What is the Binomial Theorem in summation form? _________________________

Explain how finding a term in an expansion can be used to answer a particular kind of probability question. _____________________________________________________________________________________________________________________________________________

.

1. Expand: (3x+a)5\begin{align*}(3x+a)^{5}\end{align*}
2. Expand: (xy)7\begin{align*}(x - y)^7\end{align*}
3. Expand: (2x+3)4\begin{align*}(2x + 3)^4\end{align*}
4. Find the 5th term in the expansion (4x3a)9\begin{align*}(4x-3a)^{9}\end{align*} .
5. What is the coefficient of x5\begin{align*}x^5\end{align*} in the expansion of (3x+4)6\begin{align*}(3x + 4) ^6\end{align*} ?