Quiz 9-1

1
Quiz 9-1
Permutation or Combination?
1.

• 10 person committees formed from a group of 20
  people

2.
1st, 2nd, and 3rd place trophies awarded to the
top three contestants of 100 entrants.
How many possibilities are there?
3. Pres, Vice Pres, and Secretary chosen out of
7 candidates.

• The county chooses 5 trucks from among ten
  possibilities for
• its building inspectors to use.

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HOMEWORK

• Section 9-2
• (page 715)
• (evens) 2-22, 38, 40

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9-2

• Binomial Theorem

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What youll learn about

• Powers of Binomials
• Pascals Triangle
• The Binomial Theorem
• and why
• The Binomial Theorem is a marvelous study in
• combinatorial patterns. Hopefully you will again
  marvel at the beauty of mathematics!

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Powers of Binomials

• Many important mathematical discoveries have
  begun with the study of patterns.
• We will set the stage by observing some patterns.

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Look for a pattern
1
The sum of the exponents of each term is the
same as the exponent of the binomial.
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Look for a pattern
1
The coefficients of the outer terms are always
1
8
Look for a pattern
1
9
Look for a pattern
1
10
Look for a pattern
1
11
Look for a pattern
1
12
Look for a pattern
1
There is a triangle pattern. Add the two
coefficients of the previous row to get the
coefficient of the next row.
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Pascals Triangle
This pattern was found in a Chinese text from
1303 by Chu Shih-cheh called diagram of the
old method of finding the eighth and lower
powers.

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1

When it was introduced in Europe, Blaise
Pascal used it extensively, from whom it
derived its name.
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Your turn

• Find the coefficients of the binomial
• expansion of

Exponent

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 5 10 10 5 1
• 1 6 15 20 15 6 1
• 1 7 21 35 35 21 7 1

0. 1 2. 3 4
.. 5. 6 7..
1 8 28 56 70 56 28 8 1

15
Look for a pattern
1
16
Look for a pattern
1
17
Look for a pattern
1
The number of terms in the expanded polynomial
equals the exponent of the binomial plus one.
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Vocabulary
Binomial coefficient the coefficient of each
of the individual terms of the expanded
polynomial when a binomial ? (a b) is raised
to some power.
The way to calculate the value of each of the
binomial coefficients is
Another way of writing is
(spoken n choose r)
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Finding the binomial expansion coefficients
The way to calculate each of the binomial
coefficients of the expansion of
are the values of
for r values of 0, 1, 2, 3, , n
So for n 5, the r values are 0, 1, 2, 3, 4,
5
If n 7, there are (n 1) terms
6 terms!!
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Your turn

• Find the 6th coefficient from the left of the
• binomial expansion of the binomial of

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Using nCr to Expand a Binomial
Using the TI-83 to find coefficients for
(n 4)
n choose r

Button strokes
4
4 choose
2nd
0
(accesses the catalog of functions)
log
(moves to the n section of the alphabetical
list)
enter
Now scroll down to the function
Now enter
0,1,2,3,4
(use the correct brackets)
then
enter
1 4 6 4 1
You should get
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Your turn

• Use you TI-83 to find the coefficients of
• the binomial expansion of

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What about?

• 1
• 1 -1
• 1 -2 1
• 1 -3 3 -1
• 1 -4 6 -4 1
• 1 - 5 10 -10 5 -1

Whats the pattern?
The numbers are the same!! Just the
sign of the numbers has
changed.
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The Binomial Theorem

(Pascals tri.)
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Computing Binomial Coefficients

• Find the coefficient of x13 in the expansion of
  (x 2)15.
• The only term in the expansion that we need to
  deal with is 15C2x1322

Your turn

• Calculate the entire term which has
• as the variable.

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Expanding a Binomial

• Expand (2x y2)4.
• We use the Binomial Theorem to expand (a b)4
  where a 2x and b -y2

Your turn
5. Calculate the 2nd term of the expansion
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Basic Factorial Identities

4! 4 3! 4321