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# Binomial Theorem and Expansions

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

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Practice Binomial Theorem and Expansions
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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 $n! =$ ___________________________________

Simplify and Evaluate the Factorials:

1. $\frac{(a + 3)!}{(a + 4)!}$
2. $\frac{5!}{4! 2!}$
3. $\frac{11!}{7!}$

Solve

1. $_{9}C_{7}$
2. $_{5}C_{2}$
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?

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#### Binomial Theorem and Expansions

Pascal's Triangle can help expand binomials.

It follows the pattern $\binom{n} {r - 1} + \binom{n} {r} = \binom{n + 1} {r}$ .

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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. _____________________________________________________________________________________________________________________________________________

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1. Expand: $(3x+a)^{5}$
2. Expand: $(x - y)^7$
3. Expand: $(2x + 3)^4$
4. Find the 5th term in the expansion $(4x-3a)^{9}$ .
5. What is the coefficient of $x^5$ in the expansion of $(3x + 4) ^6$ ?