More about the binomial theorem.

1. In how many ways can three officers — President, Vice President, and
Secretary — be selected from a club of ten members?
How many distinguishable permutations are there of the letters in the word
CALCULUS?
How many ways can 7 of King Arthur’s Knights be seated around the
round table?

2. How many ways can four married couples sit on a park bench if:
(a) every husband and wife must sit together?
(b) the men and women must alternate?

3. Simplify each of the following expressions:

4. The Binomial Theorem ...
Combinatorically
Algebraically
Notice the patterns ...
(1) The coefficient of the term is:
(2) The exponent on a is given by: [n - (i - 1)]
(3) The exponent on b is given by: i
(4) This relation holds for each term in the expansion:
[exponent on a] + [exponent on b] = n
(5) The number of terms in any binomial expansion is: n + 1

6. Any individual term, let's say the ith term, in a binomial expansion can be
represented like this:
Example: Find the 4th term in the expansion of

7. Recall: This relation holds for each term in any binomial expansion:
[exponent on a] + [exponent on b] = n
And any individual term in a binomial expansion can be represented like this:
Example: Find the term that contains x in the expansion of

8. Example: Find the term that contains x in the expansion of

9. Example: Find the term that contains x in the expansion of

10. Determine the indicated term in each expansion.
(a) the 8th term in the
expansion of

11. Determine the indicated term in each expansion.
(b) the 4th term in the
expansion of

12. Determine the indicated term in each expansion.
(c) the middle term in the
expansion of

13. Find the term that contains in the expansion of: