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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


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!}


  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} .


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}
  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 ?

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