Looks like we are going to need to use the definition of a **binomial
coefficient** to help us out on this.

**Looking at the definition of binomial coefficient,
what is n?**

If you said 15, you are correct!!! *n* is the top number, which in this case is 15.

**Looking at the definition of binomial coefficient,
what is r?**

If you said 2, give yourself a pat on the back!!!! *r* is the bottom number, which in this case is 2.

**Putting those values into the definition of a binomial coefficient
we get:**

*** n = 15, r = 2**

***Eval. inside ( )**

***Expand 15! until it gets to 13!**

** which is the larger ! in the den.**

***Cancel out 13!'s**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**The final answer is 105. **

Looks like we are going to need to use the definition of a **binomial
coefficient** to help us out on this.

**Looking at the definition of binomial coefficient,
what is n?**

If you said 12, you are correct!!! *n* is the top number, which in this case is 12.

**Looking at the definition of binomial coefficient,
what is r?**

If you said 12, give yourself a pat on the back!!!! *r* is the bottom number, which in this case is 12.

**Putting those values into the definition of a binomial coefficient
we get:**

***Eval. inside ( )**

***0! = 1**

***Cancel out 12!'s**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**The final answer is 1. **

Looks like we are going to need to use the **Binomial
Theorem** to help us out on this.

**Looking at the Binomial Theorem, what is a?**

If you said 2*x*, you are correct!!! *a* is the first term of the binomial, which in this case is 2*x*.

**Looking at the Binomial Theorem, what is b?**

If you said 9, give yourself a pat on the back!!!! *b* is the second term of the binomial, which in this case is 9.

**Looking at the Binomial Theorem, what is n?**

If you said 4, give yourself a high five!!!! *n* is the exponent on the binomial, which in this case is 4.

**Putting those values into the Binomial Theorem we get:**

*** a = 2x, b = 9,n = 4**

***Use definition of binomial
coefficient**

***Eval. 2 x's
and 9's raised to exponents**

***Eval. inside ( )**

***Expand num. until it gets to**

**larger ! in the den.**

***Cancel out !'s**

***Simplify**

If you don't have a factorial key, you can simplify it as shown above and then enter it in. It is probably best to simplify it first, because in some cases the numbers can get rather large, and it would be cumbersome to multiply all those numbers one by one.

**So our final answer is **.

Looks like we are going to need to use the **Binomial
Theorem** to help us out on this.

**Looking at the Binomial Theorem, what is a?**

If you said ,
you are correct!!! *a* is the first term of the
binomial, which in this case is .

**Looking at the Binomial Theorem, what is b?**

If you said ,
give yourself a pat on the back!!!! *b* is the second term of the binomial, which in this case is .

**Looking at the Binomial Theorem, what is n?**

If you said 5, give yourself a high five!!!! *n* is the exponent on the binomial, which in this case is 5.

**Putting those values into the Binomial Theorem we get:**

*** a = x^4, b = -y^6, n = 5**

***Use definition of binomial
coefficient**

***Eval. x^4's
and -y^6's raised to exponents**

***Eval. inside ( )**

***Expand num. until it gets to**

**larger ! in the den.**

***Cancel out !'s**

***Simplify**

**So our final answer is **.

; fifth
term

Looks like we are going to need to use the *r*th
term expansion formula to help us out on this.

**Looking at the rth
term expansion formula, what is n?**

If you said 8, you are correct!!! *n* is the exponent on your binomial, which in this case is 8.

**Looking at the rth
term expansion formula, what is r?**

If you said 5, give your self a pat on the back!!!! *r* is the number of the term to be found, which in this case is 5.

**Looking at the rth
term expansion formula, what is a?**

If you said ,
you are correct!!! *a* is the first term
of the binomial, which in this case is .

**Looking at the rth
term expansion formula, what is b?**

If you said 2*y*, give yourself a pat on the
back!!!! *b* is the second term of
the binomial, which in this case is 2*y*.

**Putting those values into the rth term
expansion formula we get:**

***Use definition of binomial
coefficient**

***Eval. inside ( )**

***Expand 8! until it gets to 4!**

**which is the larger ! in the den.**

***Cancel out !'s**

***Simplify**

This would tell us that the 5th term of the binomial would be .

;
fourth term

Looks like we are going to need to use the *r*th
term expansion formula to help us out on this.

**Looking at the rth
term expansion formula, what is n?**

If you said 10, you are correct!!! *n* is the exponent on your binomial, which in this case is 10.

**Looking at the rth
term expansion formula, what is r?**

If you said 4, give your self a pat on the back!!!! *r* is the number of the term to be found, which in this case is 4.

**Looking at the rth
term expansion formula, what is a?**

If you said ,
you are correct!!! *a* is the first term
of the binomial, which in this case is .

**Looking at the rth
term expansion formula, what is b?**

If you said -1/4, give yourself a pat on the back!!!! *b* is the second term of the binomial, which in this case is -1/4.

**Putting those values into the rth term
expansion formula we get:**

***Use definition of binomial
coefficient**

***Eval. inside ( )**

***Expand 10! until it gets to 7!**

**which is the larger ! in the den.**

***Cancel out !'s**

***Simplify**

This would tell us that the 4th term of the binomial would be .

Last revised on May 16, 2011 by Kim Seward.

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