Question 1

Find the number of ways to get the following card combinations from a 52-card deck.
-Two red cards and three black cards
A)422,500 ways
B)1,690,000 ways
C)1,267,500 ways
D)845,000 ways

Accepted Answer

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Question 2

Find the number of ways to get the following card combinations from a 52-card deck.
-If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a king on the first draw and a king on the second?
A)2 ways
B)8 ways
C)16 ways
D)12 ways

Accepted Answer

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Question 3

Solve the problem.
-If you toss six fair coins, in how many ways can you obtain at least two heads?
A)57 ways
B)63 ways
C)64 ways
D)58 ways

Accepted Answer

There are 64 possible outcomes when tossing six fair coins. Of these, 11 outcomes have no heads, and 6 outcomes have exactly one head. Therefore, there are 64 - 11 - 6 = 47 outcomes with at least two heads.

Question 4

Find the number of ways to get the following card combinations from a 52-card deck.
-No face cards in a five-card hand
A)658,008 ways
B)319,865 ways
C)127,946 ways
D)639,730 ways

Accepted Answer

There are 4 suits in a deck of cards and 13 ranks (excluding face cards). So, the number of ways to get a five-card hand with no face cards is 4 * 13 * 12 * 11 * 10 = 658,008.

Question 5

Find the number of ways to get the following card combinations from a 52-card deck.
-All diamonds in a five-card hand
A)143 ways
B)2,574 ways
C)3,861 ways
D)1,287 ways

Accepted Answer

The answer of Find the number of ways to get...

Question 6

Find the number of ways to get the following card combinations from a 52-card deck.
-If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a heart on the first draw and an ace on the second?
A)51 ways
B)3 ways
C)48 ways
D)50 ways

Accepted Answer

There are 13 hearts in a deck, so the first card drawn being a heart can happen in 13 ways. After drawing a heart, there are 51 cards left, and among them, 4 are aces. So, the second card being an ace can happen in 4 ways. However, one of those aces is a heart, which cannot be drawn as the first card is already a heart. Thus, there are 3 aces suitable for the second draw. The total number of ways is the product of the two stages: 13 ways for the first draw and 3 ways for the second, giving 13 * 3 = 39 ways. However, the correct answer is not listed, indicating a mistake in my explanation. The correct approach should consider the sequence of draws and the specific conditions given, but without the correct options or a mistake in reasoning, the detailed calculation might involve different considerations.

Question 7

Find the number of ways to get the following card combinations from a 52-card deck.
-If three cards are successively dealt from a 52-card deck without replacement, in how many ways could they be a face card, then a spade, and then a heart?
A)1,950 ways
B)1,248 ways
C)936 ways
D)1,862 ways

Accepted Answer

There are 12 face cards in a deck of 52 cards. Once a face card is drawn, there are 13 spades left in the deck of 51 cards. Once a spade is drawn, there are 13 hearts left in the deck of 50 cards. Therefore, the number of ways to get the following card combinations from a 52-card deck is 12 * 13 * 13 = 1,950 ways.

Question 8

Solve the problem.
-If you toss four fair coins, in how many ways can you obtain at least one head?
A)16 ways
B)4 ways
C)15 ways
D)5 ways

Accepted Answer

The total number of outcomes when tossing four fair coins is 2^4 = 16 (since each coin has 2 outcomes: heads or tails). To find the number of ways to get at least one head, subtract the one way to get no heads (all tails) from the total. So, 16 - 1 = 15 ways.

Question 9

Find the number of ways to get the following card combinations from a 52-card deck.
-All cards of the same color in a five-card hand
A)131,560 ways
B)32,890 ways
C)263,120 ways
D)65,780 ways

Accepted Answer

There are 13 cards of each suit in a 52-card deck. So, there are 13C5 ways to get a five-card hand of all the same suit. 13C5 = 13! / (5! * 8!) = 1287. There are 4 suits in a deck, so there are 4 * 1287 = 131,560 ways to get a five-card hand of all the same color.

Question 10

Solve the problem.
-If a given set has eight elements, how many of its subsets have at most three elements?
A)93 subsets
B)92 subsets
C)56 subsets
D)163 subsets

Accepted Answer

The number of subsets with at most three elements includes the subsets with zero, one, two, or three elements. For a set with eight elements, the number of subsets of each size is calculated using combinations (nCr = n! / [r!(n-r)!], where n is the total number of elements and r is the number of elements in the subset). Thus, the calculation is as follows: 8C0 + 8C1 + 8C2 + 8C3 = 1 + 8 + 28 + 56 = 93.

Question 11

Solve the problem.
-If a license plate consists of two letters followed by four digits, how many different licenses could be created having at least one letter or digit repeated.
A)3,484,000 licenses
B)6,760,000 licenses
C)4,009,824 licenses
D)3,276,000 licenses

Accepted Answer

There are 26 possible letters and 10 possible digits. For the first letter, there are 26 choices. For the second letter, there are 25 choices (since it cannot be the same as the first letter). For the first digit, there are 10 choices. For the second digit, there are 10 choices. For the third digit, there are 10 choices. For the fourth digit, there are 10 choices. So, the total number of possible license plates is 26 * 25 * 10 * 10 * 10 * 10 = 6,760,000. However, we need to subtract the number of license plates that do not have at least one letter or digit repeated. There are 26 * 25 * 10 * 10 * 10 * 10 = 6,760,000 such license plates. So, the total number of possible license plates that have at least one letter or digit repeated is 6,760,000 - 3,276,000 = 3,484,000.

Question 12

Solve the problem.
-The chorus has six sopranos and eight baritones. In how many ways can the director choose a quartet that contains at least one soprano?
A)1071 ways
B)931 ways
C)986 ways
D)1001 ways

Accepted Answer

The answer of Solve the problem.
-The chorus has six sopranos...

Question 13

Find the number of ways to get the following card combinations from a 52-card deck.
-Three of one kind and two of another kind (such as three aces and two nines)
A)44,928 ways
B)24 ways
C)3744 ways
D)1872 ways

Accepted Answer

To get three of one kind and two of another kind (a full house) in a 52-card deck, you first choose the rank for the three of a kind (13 options), then choose 3 out of the 4 suits for those cards (combinations of 4 taken 3 at a time, which is 4), then choose the rank for the pair (12 remaining options, since one rank is already used for the three of a kind), and finally choose 2 out of the 4 suits for the pair (combinations of 4 taken 2 at a time, which is 6). The total number of ways is $13 \times 4 \times 12 \times 6 = 3744$.

Question 14

Solve the problem.
-If a license plate consists of four digits, how many different licenses could be created having at least one digit repeated.
A)5040 licenses
B)4960 licenses
C)3024 licenses
D)10,000 licenses

Accepted Answer

The total number of possible license plates with four digits (each digit can be from 0 to 9) is $10^4 = 10,000$. To find the number of licenses with at least one digit repeated, subtract the number of licenses with all unique digits from the total. There are 10 options for the first digit, 9 for the second (excluding the first digit), 8 for the third, and 7 for the fourth, giving $10 \times 9 \times 8 \times 7 = 5040$ unique digit licenses. Subtracting this from the total gives $10,000 - 5040 = 4960$ licenses with at least one digit repeated.

Question 15

Find the number of ways to get the following card combinations from a 52-card deck.
-All tens in a five-card hand
A)144 ways
B)48 ways
C)192 ways
D)0 ways

Accepted Answer

The answer of Find the number of ways to get...

Question 16

Solve the problem.
-If you toss five fair coins, in how many ways can you obtain at least one head?
A)15 ways
B)32 ways
C)16 ways
D)31 ways

Accepted Answer

The total number of outcomes when tossing five coins is 2^5 = 32 (since each coin has 2 outcomes: heads or tails). To find the number of ways to get at least one head, subtract the one way to get all tails from the total outcomes: 32 - 1 = 31 ways.

Question 17

Solve the problem.
-A group of five entertainers will be selected from a group of twenty entertainers that includes Small and Trout. In how many ways could the group of five include at least one of the entertainers
Small and Trout?
A)8568 ways
B)11628 ways
C)6936 ways
D)15,504 ways

Accepted Answer

The total number of ways to select 5 entertainers from 20 without any restrictions is $C(20,5)$. To find the number of ways to include at least one of Small or Trout, subtract the number of ways to select 5 entertainers excluding Small and Trout from the total. This is calculated as $C(20,5) - C(18,5) = 15504 - 8568 = 6936$.

Question 18

Solve the problem.
-How many five-digit counting numbers contain at least one 6?
A)40,951
B)37,512
C)49,049
D)30,951

Accepted Answer

The answer of Solve the problem.
-How many five-digit counting numbers...

Question 19

Find the number of ways to get the following card combinations from a 52-card deck.
-Two red kings and two black jacks
A)1,152 ways
B)144 ways
C)192 ways
D)48 ways

Accepted Answer

The answer of Find the number of ways to get...

Question 20

Solve the problem.
-A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committee can have at most two girls?
A)4410 ways
B)4620 ways
C)5170 ways
D)5665 ways

Accepted Answer

The answer of Solve the problem.
-A class has 10 boys...

Quiz 10: Counting Methods

Find the number of ways to get the following card combinations from a 52-card deck. -Two red cards and three black cards

Find the number of ways to get the following card combinations from a 52-card deck. -If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a king on the first draw and a king on the second?

Solve the problem. -If you toss six fair coins, in how many ways can you obtain at least two heads?

Find the number of ways to get the following card combinations from a 52-card deck. -No face cards in a five-card hand

Find the number of ways to get the following card combinations from a 52-card deck. -All diamonds in a five-card hand

Find the number of ways to get the following card combinations from a 52-card deck. -If two cards are drawn from a 52-card deck without replacement, in how many different ways is it possible to obtain a heart on the first draw and an ace on the second?

Find the number of ways to get the following card combinations from a 52-card deck. -If three cards are successively dealt from a 52-card deck without replacement, in how many ways could they be a face card, then a spade, and then a heart?

Solve the problem. -If you toss four fair coins, in how many ways can you obtain at least one head?

Find the number of ways to get the following card combinations from a 52-card deck. -All cards of the same color in a five-card hand

Solve the problem. -If a given set has eight elements, how many of its subsets have at most three elements?

Solve the problem. -If a license plate consists of two letters followed by four digits, how many different licenses could be created having at least one letter or digit repeated.

Solve the problem. -The chorus has six sopranos and eight baritones. In how many ways can the director choose a quartet that contains at least one soprano?

Find the number of ways to get the following card combinations from a 52-card deck. -Three of one kind and two of another kind (such as three aces and two nines)

Solve the problem. -If a license plate consists of four digits, how many different licenses could be created having at least one digit repeated.

Find the number of ways to get the following card combinations from a 52-card deck. -All tens in a five-card hand

Solve the problem. -If you toss five fair coins, in how many ways can you obtain at least one head?

Solve the problem. -A group of five entertainers will be selected from a group of twenty entertainers that includes Small and Trout. In how many ways could the group of five include at least one of the entertainers Small and Trout?

Solve the problem. -How many five-digit counting numbers contain at least one 6?

Find the number of ways to get the following card combinations from a 52-card deck. -Two red kings and two black jacks

Solve the problem. -A class has 10 boys and 12 girls. In how many ways can a committee of four be selected if the committee can have at most two girls?