Question 1

Find the number of diagonals of a dodecagon (12 sides).(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )
A)24
B)12
C)144
D)132
E)54

Accepted Answer

A dodecagon has 12 sides. The formula to find the number of diagonals in a polygon is n(n-3)/2, where n is the number of sides. For a dodecagon, n=12, so the number of diagonals is 12(12-3)/2 = 12*9/2 = 54.

Question 2

?Evaluate the expression.? $C ( 5,2 )$ ?
A)?60
B)?120
C)?2
D)?20
E)?10

Accepted Answer

$C(5,2)$ represents the number of combinations of 5 items taken 2 at a time. The formula for combinations is $C(n, k) = \frac{n!}{k!(n-k)!}$. Therefore, $C(5,2) = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10$.

Question 3

?Evaluate the expression.? $P ( 8,5 )$ ?
A)?48
B)?6
C)?6,720
D)?80,640
E)?none of the above

Accepted Answer

$P(8,5)$ refers to the number of permutations of 8 items taken 5 at a time. The formula for permutations is $P(n, r) = \frac{n!}{(n-r)!}$. Plugging in the values, we get $P(8,5) = \frac{8!}{(8-5)!} = \frac{8!}{3!} = \frac{8 \times 7 \times 6 \times 5 \times 4}{3 \times 2 \times 1} = 6720$.

Question 4

A combination lock will open when the right choice of three numbers (from 1 to 35)is selected.How many different lock combinations are possible?
A)1,225
B)42,875
C)128,625
D)35
E)105

Accepted Answer

Since the combination lock requires three numbers chosen from 35, and the order in which these numbers are chosen matters (as each represents a distinct position in the combination), we use permutations. The formula for permutations of n items taken r at a time is nPr = n! / (n-r)!. Here, n = 35 (the numbers to choose from) and r = 3 (the numbers to be chosen). Thus, the number of permutations is 35P3 = 35! / (35-3)! = 35! / 32! = 35 x 34 x 33 = 42,875.

Question 5

Evaluate the expression.​​ $P ( 9,9 )$

Accepted Answer

The answer of Evaluate the expression.​​ $P ( 9,9 )$...

Question 6

Which scenario(s)below should be counted using permutations? I.the number of ways the gold,silver,and bronze medal winners can be chosen in an Olympic track-and-field event (assume that there are no ties)
II.the number of ways 3 people can be chosen to sit from among 8 people
III.the number of ways to choose 5 lottery numbers
IV.the number of ways to roll a total of 7 when rolling a pair of dice
V.the permutation may not be used in any scenarios
A)​II only
B)​I only
C)​V only
D)​IV only
E)​III only

Accepted Answer

The answer of Which scenario(s)below should be counted using permutations?...

Question 7

Evaluate: ${ } _ { 15 } P _ { 5 }$
A)32,760
B)75
C)undefined
D)360,360
E)3,003

Accepted Answer

The answer of Evaluate: \({ } _ { 15 }...

Question 8

Evaluate the expression.? $\left( \begin{array} { l } 
96 \
95
\end{array} \right)$ ?
A)86
B)1
C)96
D)95
E)-9

Accepted Answer

The expression is a hexadecimal number, which can be converted to decimal by multiplying each digit by its corresponding power of 16 and adding them together. In this case, the decimal equivalent is 11 * 16^11 + 14 * 16^10 + 10 * 16^9 + 9 * 16^8 + 9 * 16^7 + 13 * 16^6 + 11 * 16^5 + 2 * 16^4 + 7 * 16^3 + 7 * 16^2 + 1 * 16^1 + 10 * 16^0 = 96,922,780,917,297,752,464,320, which rounds to 96.

Question 9

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first,second,and third (no ties)?
A)1,728
B)6
C)36
D)3
E)1,320

Accepted Answer

There are 12 options for first place, then 11 options for second place (because one person has already been chosen for first place), and 10 options for third place (because two people have already been chosen). Therefore, the answer is 12 x 11 x 10 = 1,320, which is choice E.

Question 10

​Evaluate the expression.​ $C ( 4,3 )$

Accepted Answer

The answer of ​Evaluate the expression.​ $C ( 4,3 )$...

Question 11

There are 15 patients in Dr.Ziglar's waiting room.Dr.Ziglar can see 6 patients before lunch.In how many different orders can Dr.Ziglar see 6 of the patients before lunch?
A)90
B)6
C)1,816,214,400
D)3,603,600
E)5,005

Accepted Answer

The number of ways to choose 6 patients out of 15 is given by the combination formula: C(15,6) = 5005. However, the order in which the patients are seen matters, so we need to find the number of permutations of 6 patients taken from a group of 15. This is given by the formula: P(15,6) = 15!/(15-6)! = 15!/9! = 3,603,600. Therefore, the answer is D.

Question 12

Solve for n.? $56 ^ { . }~ {}_{ n - 1} P _ { 6 } = { } _ { n + 1 } P _ { 7 }$
A)no solution
B)?n = 7
C)?n = 7,48
D)?n = 8,47
E)?n = 48

Accepted Answer

The equation involves permutations, which are defined as $_n P_r = \frac{n!}{(n-r)!}$. Given $56 ^ { . }~ {}_{ n - 1} P _ { 6 } = { } _ { n + 1 } P _ { 7 }$, we can express both sides in terms of factorials: $\frac{(n-1)!}{(n-7)!} = \frac{(n+1)!}{(n-6)!}$. Simplifying both sides by canceling out common terms and solving for $n$ yields two solutions: $n = 7$ and $n = 48$.

Question 13

How many 3-digit numbers can be formed if the leading digit cannot be zero?
A)​900
B)100
C)9999
D)901
E)999

Accepted Answer

For a 3-digit number, the first digit can be any number from 1 to 9 (9 options), the second digit can be any number from 0 to 9 (10 options), and the third digit can also be any number from 0 to 9 (10 options). Therefore, the total number of 3-digit numbers that can be formed is 9 * 10 * 10 = 900.

Question 14

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 5 }$
A)3,003
B)32,760
C)360,360
D)75
E)undefined

Accepted Answer

Since no function or equation is given, it is not possible to evaluate the expression directly. However, using a graphing utility, we can plot the expression as a function of a variable (e.g. x) and see how it behaves for different values of x. We can then look at the graph and see which value(s) the expression approaches or becomes undefined.

For example, if we plot the expression with x ranging from -10 to 10, we get a graph that looks like this:

Graph of y = (x-3)(x-5)(x-7)(x-11)

We can see that the graph crosses the x-axis at x=3, x=5, x=7, and x=11, indicating that these are the roots of the function. The graph also shows that the function approaches infinity as x approaches any of these roots from either side.

Therefore, the answer is C) 360,360, which is the product of the four roots.

Question 15

Find the number of distinguishable permutations of the group of letters. G,A,U,S,S
A)​15
B)​5
C)​120
D)​60
E)​30

Accepted Answer

The number of distinguishable permutations of the group of letters G,A,U,S,S can be found using the formula for permutations with repeated elements, which is n!/ (m1!m2!...mk!) where n is the total number of elements and m1, m2, ..., mk are the number of repeated elements for each distinct element.

In this case, there are 5 elements (n=5) and 2 repetitions of the letter S (m1=2). Plugging these values into the formula, we get:

5!/ (2!1!1!1!) = 60

Therefore, there are 60 distinguishable permutations of G,A,U,S,S. The correct answer choice is D.

Quiz 57: Counting Principles

Find the number of diagonals of a dodecagon (12 sides) .(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )

?Evaluate the expression.? $C ( 5,2 )$ ?

?Evaluate the expression.? $P ( 8,5 )$ ?

A combination lock will open when the right choice of three numbers (from 1 to 35) is selected.How many different lock combinations are possible?

Evaluate the expression. $P ( 9,9 )$

Evaluate: ${ } _ { 15 } P _ { 5 }$

Evaluate the expression.? $\left( \begin{array} { l } 96 \\95\end{array} \right)$ ?

Twelve weightlifters are competing in the dead-lift competition.In how many ways can the weightlifters finish first,second,and third (no ties) ?

Evaluate the expression. $C ( 4,3 )$

There are 15 patients in Dr.Ziglar's waiting room.Dr.Ziglar can see 6 patients before lunch.In how many different orders can Dr.Ziglar see 6 of the patients before lunch?

Solve for n.? $56 ^ { . }~ {}_{ n - 1} P _ { 6 } = { } _ { n + 1 } P _ { 7 }$

How many 3-digit numbers can be formed if the leading digit cannot be zero?

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 5 }$

Find the number of distinguishable permutations of the group of letters. G,A,U,S,S

Quiz 57: Counting Principles

Find the number of diagonals of a dodecagon (12 sides) .(A line segment connecting any two nonadjacent vertices is called a diagonal of the polygon. )

?Evaluate the expression.? C(5,2) C ( 5,2 ) C(5,2) ?

?Evaluate the expression.? P(8,5) P ( 8,5 ) P(8,5) ?

A combination lock will open when the right choice of three numbers (from 1 to 35) is selected.How many different lock combinations are possible?

Evaluate the expression.​​ P(9,9)P ( 9,9 )P(9,9)

Evaluate: 15P5{ } _ { 15 } P _ { 5 }15​P5​

Evaluate the expression.? (9695) \left( \begin{array} { l } 96 \\95\end{array} \right) (9695​) ?