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Binomial Theorem and Expansions

Expansion of binomials raised to a power using combinations.

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

Simplify and Evaluate the Factorials:

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

Solve

1. \begin{align*} _{9}C_{7}\end{align*}
2. \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?

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Binomial Theorem and Expansions

Pascal's Triangle can help expand binomials.

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

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