Question 1

Evaluate   $C ( 9,6 )$ .

Accepted Answer

The combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$ is used to calculate $C(9, 6)$. This equals $\frac{9!}{6!3!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84$.

Question 2

The "Dogs of the Dow" refer to the stocks listed on the Dow with the highest dividend yield. The table shows the top ten stocks of the "Dogs of the Dow" list in February, 2005. You decide to make a small portfolio consisting of a collection of seven of the top ten Dogs of the Dow. How many portfolios are possible  $$\begin{array} { | c | c | c | c | } 
\hline \text { Symbol } & \text { Company } & \text { Price } & \text { Yield } \
\hline \text { SBC } & \text { SBC } & 23.62 & 5.46 \% \
\hline \text { Communications } & \text { General Motors } & 36.64 & 5.46 \% \
\hline \text { MRK } & \text { Merck } & 28.02 & 5.42 \% \
\hline \text { MO } & \text { Altria } & 63.20 & 4.62 \% \
\hline \text { VZ } & \text { Verizon } & 35.68 & 4.32 \% \
\hline \text { JPM } & \text { JP Morgan Chase } & 37.00 & 3.68 \% \
\hline \text { C } & \text { Citigroup } & 48.38 & 3.64 \% \
\hline \text { PFE } & \text { Pfizer } & 24.35 & 3.12 \% \
\hline \text { DD } & \text { DuPont } & 47.23 & 2.96 \% \
\hline \text { GE } & \text { General Electric } & 35.75 & 2.46 \% \
\hline
\end{array}$$

Accepted Answer

D

Question 3

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 4 marbles include at most one of the yellow marbles

Accepted Answer

D

Question 4

The number of ordered lists of 5 different items, chosen 8 at a time is 56.

Accepted Answer

The number of ordered lists of 5 different items chosen 8 at a time would be much greater than 56, as there are more than 8 different ways to choose 5 items out of 8. Therefore, the statement is false.

Question 5

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 3 marbles include none of the yellow marbles

Accepted Answer

There are 11 marbles that are not yellow. To choose 3 of these marbles, there are 11C3 = 165 ways. However, we have overcounted the cases where the 3 marbles are all red, all blue, or all lavender. There are 3C3 = 1 way to choose 3 red marbles, 6C3 = 20 ways to choose 3 blue marbles, and 3C3 = 1 way to choose 3 lavender marbles. Therefore, the number of sets of 3 marbles that include none of the yellow marbles is 165 - 1 - 20 - 1 = 143.

Question 6

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of four of a kind (all four of one denomination and one of another). ​

Accepted Answer

To form a "four of a kind" hand, choose 1 of the 13 denominations for the four cards, and then choose 1 of the remaining 48 cards for the fifth card. This gives 13 * 48 = 624 different hands.

Question 7

Evaluate   $C ( 7,4 )$ .

Accepted Answer

$C(7,4) = \frac{7!}{4!(7-4)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35.$

Question 8

Evaluate  $P ( 8,5 )$ .

Accepted Answer

$P(8,5)$ refers to the number of permutations of 8 items taken 5 at a time. The formula for permutations is $P(n, k) = \frac{n!}{(n-k)!}$. 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 9

If 8 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes  ?

Accepted Answer

The total number of handshakes can be calculated using the formula for combinations: C(n, 2) = n! / [2!(n-2)!], where n is the total number of people. For 8 people, it's C(8, 2) = 8! / [2!(8-2)!] = 28 handshakes.

Question 10

How many ordered sequences are possible that contain 2 objects chosen from 9

Accepted Answer

The number of ordered sequences of 2 objects chosen from 9 is calculated using permutations, given by 9P2 = 9! / (9-2)! = 9*8 = 72.

Question 11

The number of ordered lists of 7 different items, chosen 4 at a time is 840.

Accepted Answer

The number of ordered lists of 7 different items chosen 4 at a time is calculated using permutations: $P(n, k) = \frac{n!}{(n-k)!}$. For $n=7$ and $k=4$, it is $\frac{7!}{(7-4)!} = \frac{7!}{3!} = \frac{5040}{6} = 840$.

Question 12

How many ordered lists are there of 4 items chosen from 6  ?

Accepted Answer

The number of ordered lists of 4 items chosen from 6 is calculated using permutations since the order matters. The formula for permutations is $P(n, r) = \frac{n!}{(n-r)!}$, where $n$ is the total number of items to choose from, and $r$ is the number of items to choose. Here, $n = 6$ and $r = 4$, so $P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!} = \frac{720}{2} = 360$.

Question 13

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 1 lavender marble. How many possible sets of three marbles are ther

Accepted Answer

The total number of marbles is 3 + 6 + 2 + 1 = 12. The number of ways to choose 3 marbles out of 12, without regard to order, is calculated using the combination formula C(n, k) = n! / [k!(n - k)!], where n is the total number of items, k is the number of items to choose, and "!" denotes factorial. Thus, C(12, 3) = 12! / [3!(12 - 3)!] = 12! / (3!9!) = (12 × 11 × 10) / (3 × 2 × 1) = 220.

Question 14

Find the indicated value. ​7!
​

Accepted Answer

7! (7 factorial) means 7 × 6 × 5 × 4 × 3 × 2 × 1, which equals 5,040.

Question 15

How many eight-letter sequences are possible that use the letters f, i, t, t, i, n, g, s

Accepted Answer

The sequence has 8 letters with "t" and "i" each appearing twice. The formula for permutations of a multiset is $n! / (n_1! \cdot n_2! \cdot ... \cdot n_k!)$, where $n$ is the total number of items, and $n_1, n_2, ..., n_k$ are the counts of each unique item. Here, $n = 8$, $n_1 = n_2 = 2$ (for "t" and "i"), so the number of sequences is $8! / (2! \cdot 2!) = 40,320 / 4 = 10,080$.

Question 16

Evaluate   $P ( 7,5 )$ .

Accepted Answer

The permutation formula $P(n, r) = \frac{n!}{(n-r)!}$ is used here. For $P(7, 5)$, it is $\frac{7!}{(7-5)!} = \frac{7!}{2!} = \frac{5040}{2} = 2520$.

Question 17

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 36 cards

Accepted Answer

The answer of In how many ways can a 6-card...

Question 18

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A). ​

Accepted Answer

To calculate the number of different flush hands (excluding straight flushes and royal flushes), we first choose a suit (4 ways), and then choose 5 cards out of the 13 available in that suit. The number of ways to choose 5 cards from 13 is given by the combination formula $C(n, k) = \frac{n!}{k!(n-k)!}$, where $n$ is the total number of items, $k$ is the number of items to choose, and "!" denotes factorial, the product of all positive integers up to that number. Thus, the calculation is $4 \times C(13, 5) = 4 \times \frac{13!}{5!(13-5)!} = 4 \times \frac{13!}{5!8!} = 4 \times 1,287 = 5,148$.

Question 19

Evaluate $\frac { 8 ! } { 7 ! }$ .

Accepted Answer

The expression $\frac{8!}{7!}$ simplifies to $\frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$, which further simplifies to 8, because all the terms except 8 cancel out.

Question 20

How many six-letter sequences are possible that use the letters g, o, b, l, i, n once each  ?

Accepted Answer

There are 6 choices for the first letter, 5 choices for the second letter, 4 choices for the third letter, and so on until there is only 1 choice left for the last letter. Therefore, the total number of sequences is 6 x 5 x 4 x 3 x 2 x 1 = 720.

Quiz 7: Sets and Counting

Evaluate $C ( 9,6 )$ .

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 4 marbles include at most one of the yellow marbles

The number of ordered lists of 5 different items, chosen 8 at a time is 56.

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 3 marbles include none of the yellow marbles

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of four of a kind (all four of one denomination and one of another) .

Evaluate $C ( 7,4 )$ .

Evaluate $P ( 8,5 )$ .

If 8 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes ?

How many ordered sequences are possible that contain 2 objects chosen from 9

The number of ordered lists of 7 different items, chosen 4 at a time is 840.

How many ordered lists are there of 4 items chosen from 6 ?

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 1 lavender marble. How many possible sets of three marbles are ther

Find the indicated value. 7!

How many eight-letter sequences are possible that use the letters f, i, t, t, i, n, g, s

Evaluate $P ( 7,5 )$ .

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 36 cards

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A) .

Evaluate $\frac { 8 ! } { 7 ! }$ .

How many six-letter sequences are possible that use the letters g, o, b, l, i, n once each ?

Quiz 7: Sets and Counting

Evaluate C(9,6) C ( 9,6 ) C(9,6) .

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 4 marbles include at most one of the yellow marbles

The number of ordered lists of 5 different items, chosen 8 at a time is 56.

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 3 lavender marbles. How many sets of 3 marbles include none of the yellow marbles

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of four of a kind (all four of one denomination and one of another) . ​

Evaluate C(7,4) C ( 7,4 ) C(7,4) .

Evaluate P(8,5) P ( 8,5 ) P(8,5) .

If 8 persons meet at a reunion and each person shakes hands exactly once with each of the others, what is the total number of handshakes ?

How many ordered sequences are possible that contain 2 objects chosen from 9

The number of ordered lists of 7 different items, chosen 4 at a time is 840.

How many ordered lists are there of 4 items chosen from 6 ?

A bag contains 3 red marbles, 6 blue marbles, 2 yellow marbles, and 1 lavender marble. How many possible sets of three marbles are ther

Find the indicated value. ​7! ​

How many eight-letter sequences are possible that use the letters f, i, t, t, i, n, g, s

Evaluate P(7,5) P ( 7,5 ) P(7,5) .

In how many ways can a 6-card hand containing 4 kings and 2 queens be dealt from a deck of 36 cards

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A) . ​

Evaluate 8!7!\frac { 8 ! } { 7 ! }7!8!​ .

How many six-letter sequences are possible that use the letters g, o, b, l, i, n once each ?

Evaluate $C ( 9,6 )$ .

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of four of a kind (all four of one denomination and one of another) .

Evaluate $C ( 7,4 )$ .

Evaluate $P ( 8,5 )$ .

Find the indicated value. 7!

Evaluate $P ( 7,5 )$ .

A poker hand consists of five cards from a standard deck of 52. Find the number of different poker hands of the flush (five cards all of the same suit, but not consecutive denominations: A, 2, 3, 4, 5 up through 10, J, Q, K, A) .

Evaluate $\frac { 8 ! } { 7 ! }$ .