pre-cal 40s slides april 26, 2007
DESCRIPTION
More about the binomial theorem.TRANSCRIPT
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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?
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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?
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Simplify each of the following expressions:
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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
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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
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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:
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Example: Find the term that contains x in the expansion of
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Example: Find the term that contains x in the expansion of
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Determine the indicated term in each expansion.
(a) the 8th term in the
expansion of
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Determine the indicated term in each expansion.
(b) the 4th term in the
expansion of
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Determine the indicated term in each expansion.
(c) the middle term in the
expansion of
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Find the term that contains in the expansion of: