Download Permutations, Combinations and Binomial Theorem Review

Permutations, Combinations and Binomial Theorem Review
1.
On a quiz there are 5 multiple-choice questions each with 4 choices. How many ways of
answering the questions are possible?
2.
A restaurant’s menu shows a choice of 6 different dinners, 4 desserts and 5 beverages. In how
many ways can a person order a meal if a dinner, dessert, and a beverage is ordered?
3.
Simplify:
n C2 
a
c.
4.
In Pascal’s Triangle, what is 3
b
C2 3 C3 
Solve for n:
a
without a formula, find
 n  3 ! =
 n  1!
n
C20 n C30
P2  72  n1 P1 
b
n
c
 n  2 !  42
n!
d.
n C2 
P3
3!
n
LEAVE ANSWERS IN UNSIMPLIFIED FACTORIAL FORM:
5.
Using the letters of the word INTERMITTENT how many 12-letter words can be formed:
a.
without restrictions
b.
if each word begins with T and ends with N
c.
if the 4 T’s are together
6.
d.
if the T’s are in their original position
e.
if each word begins with 2 T’s
f.
if each word begins with exactly 2 T’s
Using the letters of the word LOGARITHM, how many 9-letter words can be formed:
a.
without restrictions
b.
if the letter L must be in the middle
c.
if the order LGR must be the same
d.
if the letters O, A, and I must remain in the exact same position
e.
7.
if the letters L , O , and G are not together
How many ways can 4 boys and 4 girls be arranged alternately in a row if:
a.
there are no other restrictions
b.
if a boy must be first in the row.
8.
A construction camp is equipped with 5 signal flags of different colors. How many different
signals can be sent if at most 3 flags can be flown from the flagpole at a time?
9.
How many ways can the letters of the word ENGLISH be arranged if the order of the vowels are
not changed?
10.
Six identical math texts, a chemistry, a physics, and a biology text are arranged on a shelf.
How many arrangements:
a.
are there
b.
will not have all of the math books together
11.
12.
13.
c.
will have the chemistry and physics text adjacent
d.
will have the chemistry text at the front and the biology text at the back
Using only 877799, how many even 6 digit numbers may be formed?
A dance committee consists of 4 girls and 5 boys. How many decorating committees of 6 can be
formed if a committee is to have a girl as chairman, a boy as lighting director, and exactly 3 girls
on the committee?
From 8 men and 5 women, a committee of 4 is selected. How many groups will have
a
3 women and 1 man
b.
Mary but not Roger
c.
no more than 2 men
d.
Roger and Joe
e.
at least 1 of each gender
f.
Roger or Joe
14.
Points are marked on the circumference of a circle. If 20 different triangles can be formed
using these points as vertices, what is the minimum number of points that are marked?
15.
How many different four-letter words can be formed by using 2 letters from GLAZED and 2 from
FROM?
16.
A box contains 5 red, 3 green, and 6 blue marbles. Three marbles are selected at random
without replacement. Determine the number of ways of selecting:
a.
exactly 2 blue
b.
at least 1 is blue
c.
all 3 are of the same color
17.
a.
If the number of terms in the expansion
b.
The value of n in the expansion of
 3x  5 y 
 x  y
n
2 n 3
is 16, then n is
if the 6th term is 38 760
x15 y 5 is
10
18.
Given
1


x

 , find
x


a. the general term in simplified form
b. the coefficient of
x4
c.
the middle term
d.
the number of the term containing
x4
e. the constant term
19.
Give the last three terms of the expansion of
 x2  2 y 
11
 x  y
18
20.
Find the ninth term of
21.
On each grid, how many different paths are there from A to B in the following street
arrangements.
a.
. How many terms would be in the expansion?
A
b.
B
B
c.
d
A
B
A
e.
f. You must go through B to get to C
g.
A
A
B
C
B
Determine the number of distinct pathways from A to B assuming you do not backtrack.
h.
i.
A
A
B
B
22.
Consider the first part of one row in Pascal’s triangle:
1, 18, 153, 816, 3 060, 8 568,18 564, 31 824, 43 758, 48 620, 43 758
a.
Part of which row in the triangle is illustrated?
b.
List all the other numbers in the row explaining how you determined their value.
c.
Explain your strategy for determining the value of the numbers in the row above this row.
23.
A regular polygon has 170 diagonals. How many sides does it have?
24.
A pizza shop has 5 favorite toppings. The number of different ways of selecting one or more of
these toppings is
25.
Suppose there are a certain number of points on a circle. If 120 triangles are formed by joining
any 3 of these points, how many points are on the circle?
26.
A club contains 15 members. In how many ways could
a.
an executive of 3 members be selected?
b.
a president, vice-president, and treasurer be selected?
Probability with Permutations and Combinations
27.
In the game of cribbage, 6 cards are dealt. What is the probability of
a. being dealt four 5’s
b.
at least one 5
28.
A lock on a briefcase has 3 wheels, each labeled from 0 to 9. What is the probability of a
person’s guessing the correct entry code, if
a.
there are no repeated digits in the code
b.
29.
the code may have repeated digits
A collection of 15 transistors contains 3 that are defective.
If 2 transistors are selected in succession without replacement, what is the
probability that:
a.
at least 1 of them is non-defective
b.
1 defective and 1 non-defective
c.
neither are defective
d.
30.
the first is defective and the second non-defective.
In a swim meet, there are 8 entries, 3 of whom come from the Coronation Swim Club. If we
assume that their abilities are about the same, what is the probability that:
a.
the Coronation Swim Club, Anna, Beth and Candice, will finish 1st, 2nd, and 3rd.
b.
there will be no Coronation swimmers in the top 3
31.
A bag contains 7 red and 4 white marbles. Three balls are selected. What is the
probability that the following occurs?
a. all the same color b.
at least 1 white
KEY SOLUTIONS
1.
1024
2.
120
3.
a.
b.
 n  3 n  2 n  1 n
4.
a.
b.
n  72
5.
a.
7.
a.
9!
b.
8!
d.
6!
e.
9!   7! 3!
a.
 4! 4! 2
b.
 4! 4!
10.
a.
d.
7!
6!
12.
 4 C1  5 C1  3 C2  4 C2 
e.
c.
f.
c.
4
C3
d.
9!4!
2!2!4!2!
4  3  8  9!
2!2!4!2!
9!
3!
5 1
c.
e.
14.
4 10! 2
2!4!2!2!
4  3 10!
2!2!4!2!
n5
b.
P 5 P2 5 P3
7!
9.
a.
2!
8! 2!
10. c.
6!
5!
 10
11.
3!2!
13. a.
 5 C3  8 C1 
8.
c.
c.
12!
2!2!4!2!
8!4!
2!2!4!2!
d.
6.
n  n  1
2
n  50
n
9!
6!
b.
 1C0  1C1  11C3 
d.
 5 C4  8 C0    5 C3  8 C1    5 C2  8 C2 
f.
 5 C3  8 C1    5 C2  8 C2    5 C1  8 C3 
C3  20
trial and error, n  6
9! 6! 4!

6!
6!
b.
15.
6
C2 4 C2  4!
 2 C2  11C2 
 13 C4    2 C0  11C4 
n5
16.
17.
18.
a.
 6 C2  8 C1 
c.
 5 C3    3 C3    6 C3 
a.
a.
6
C x
 6 C1  8 C2    6 C2  8 C1    6 C3  8 C0  or
14 C3   8 C3  6 C0 
b.
2 k 10
10 k
th
d.
8 term
b.
b.
20
e.
252
k  7,120 x 4
c.
252
153x y  18xy  y
20. 42 240x 6 y 8 , 12 terms (i.e. degree + 1)
21. a.
60
b.
48
c.
110
19.
22.
2
e.
a.
b.
c.
23.
n  20
26.
b.
28.
29.
30.
16
17
18
d.
30
30
f.
200
g.
322
h.
18
i.
44 100
19th row
since rows are symmetrical about its middle value and 43 758 has been repeated; therefore
the remaining numbers are 31 824, 18 564, 8 568, 3 060, 816, 153, 18 and 1.
The value of a number in a row below a given row can be determined by adding the two
numbers immediately above it, we reverse this process to determine the values in the row
above this row. E.g. The first number must be 1 and the therefore the next number must
be 17 since 1 + 17 = 18.
31
25.
10
15 3
27.
a.
0.000 055
a.
0.001 389
b.
a.
34
35
1
56
b.
0.001
12
35
5
28
a.
24.
P  2730
b.
26.
C3  455
a.
15
b.
0.397 23
c.
22
35
31.
a.
d.
13
55
b.
6
35
26
33
Written Response Questions
Name ________________
1. Use the expansions of  2 x  3 for each of the following
6

Find the 6th term

Find the coefficient of the middle term

Find the number of ways of choosing 2 or more people from 6 people using choose
notation.

State how you could use  x  y  and Pascal’s triangles to solve the question in bullet 3
6


2. Use the expansion of  2x 
10
1
 for each of the following.
x

Find the 5th term

Find the coefficient of the 8th term

Find the term containing x 4

Find the constant term
3. Use the letters in the word ENGINEER for each of the following.

Find the number of arrangements of all the letters.

How many arrangements begin with the letter g?

How any arrangements have the g and r adjacent and the g before the r?
gr

How many arrangements have the 3 E’s together?
e
e
e

enine
e
gninr
How many arrangements begin and end with an N?
4. Use the following information for the next questions.
The video store that Joe visits has 6 video games and 5 movies that he likes. He has enough
money to buy a total of 5 items ( movies and video games together).

Joe realizes he has enough money to purchase a total of 5 items and will select from the
ones he likes. How many different selections can he make?

How many ways can he select 2 video games and 3 movies?

How many ways can be select at least one movie and the rest video games?
Written Response Questions
1. Use the expansions of  2 x  3 for each of the following
6

Find the 6th term
t6 6 C 5  2 x   3
1
5
 6  2 x   243
1
 2916 x

Find the coefficient of the middle term
t4 6 C 3  2 x   3
3
t4 6 C 3  2 x   3
3
3
 20  8 x3   27 
6!
8 x3   27 

3!3!
 4320 x3
 4320

 4320 x3
 4320

3
Find the number of ways of choosing 2 or more people from 6 people using choose
notation.
6 C2  6 C3  6 C4  6 C5  6 C6
6!
6!
6!
6!
6!




2!4! 3!3! 4!2! 5!1! 6!0!
 15  20  15  6  1
 57


State how you could use  x  y  and Pascal’s triangles to solve the question in bullet 3
6
  x  y
6
the sum of the coefficients = 26  64
 64   6 C0  6 C1 
 64  7
 57
10


2. Use the expansion of  2x 

1
 for each of the following.
x
Find the 5th term
4
 1
t5 10 C 4  2 x    
 x
10!
 1 

64 x 6   4 

4!6!
x 
 1 
 210  64 x 6   4 
x 
6
 13440 x 2

Find the coefficient of the 8th term
7
 1
t8 10 C 7  2 x    
 x
10!
 1 

8 x3    7 

7!3!
 x 
3
 120  8 x 4 
 960 x 4

Find the term containing x 4
tr 1  n C r  2 x 
10  r
 1
 
 x
r
10 C r  2 
 1  x 
10 C r  2 
r
10  r
r
10  r
 1
 
 x
 1  x 
10  r
r
10  2 t
10 C r  2   1  x 
10  r
10  r  r
10  2r  4
6  2r
3r
t4 10 C 3  2 
 1  x 
7
3
4
10 C 3  2   1  x 
 120 121 1 x 4
10 3
 15360 x 4
3
10  2*3
r

Find the constant term
tr 1 10 C r  2 
10  r
 1  x 
r
10  2 t
10  2r  0
10  2r
5r
t6 10 C 5  2 
 1  x 
5
5
0
10 C 5  2   1  x 
 252  32  11
10  5
5
10  2*5
 8064
3. Use the letters in the word ENGINEER for each of the following.

Find the number of arrangements of all the letters.
8!
 3360
3!2!

How many arrangements begin with the letter g?
1 7!
3!2!

How any arrangements have the g and r adjacent and the g before the r?
gr

 3360
enine
e
6 x 5 x 4x 3 x 2 x 1
gr is a unit and can
be located in 7 positions
therefore
7!
 420
3!2!
How many arrangements have the 3 E’s together?
e
e
e
gninr
5 x 4x 3 x 2 x 1
eee’s is a unit and can
be located in 6 positions
therefore:
3!6!
 360
3!2!

How many arrangements begin and end with an N?
 2  6!1
N __ __ __ __ __ __ N
3!2!
 120
5. Use the following information for the next questions.
The video store that Joe visits has 6 video games and 5 movies that he likes. He has enough
money to buy a total of 5 items ( movies and video games together).

Joe realizes he has enough money to purchase a total of 5 items and will select from the
ones he likes. How many different selections can he make?
11 C5


11!
 462
5!6!
How many ways can he select 2 video games and 3 movies?
6!
5!

2!4! 3!2!
 15 10 
6 C2 5 C3 
 150

How many ways can be select at least one movie and the rest video games?
5 C1 6 C4 5 C2 6 C3 5 C3 6 C2 5 C4 6 C1 5 C5
 75  200  150  30  1
 456
or
11 C5   no movies 
11 C5 6 C5
 462  6
 456