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?

How many ways can four married couples sit on a park bench if:

(b) the men and women must alternate?

(a) every husband and wife must sit together?

Simplify each of the following expressions:

The Binomial Theorem ...

Algebraically

Combinatorically

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

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

Recall: This relation holds for each term in any binomial expansion:

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

[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

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

Determine the indicated term in each expansion.

(a) the 8th term in the

expansion of

Determine the indicated term in each expansion.

(b) the 4th term in the

expansion of

Determine the indicated term in each expansion.

(c) the middle term in the

expansion of

Find the term that contains in the expansion of: