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Q:

Eight first class and six second class petty officers are on the board of the 56 club. In how many ways can the members elect, from the board, a president, a vice-president, a secretary, and a treasurer if the president and secretary must be first class petty officers and the vice-president and treasurer must be second class petty officers?

A) 1500	B) 1860
C) 1680	D) 1640

Answer & Explanation Answer: C) 1680

Explanation:

Since two of the eight first class petty officers are to fill two different offices, we write $8 P_{2} =$ 56

Then, two of the six second class petty officers are to fill two different offices; thus, we write $6 P_{2}$ =30

The principle of choice holds in this case; therefore, the members have 56 x 30 = 1680 ways to select the required office holders

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Q:

A team of 8 students goes on an excursion, in two cars, of which one can seat 5 and the other only 4. In how many ways can they travel?

A) 126	B) 240
C) 120	D) 260

Answer & Explanation Answer: A) 126

Explanation:

There are 8 students and the maximum capacity of the cars together is 9.

We may divide the 8 students as follows

Case I: 5 students in the first car and 3 in the second Or

Case II: 4 students in the first car and 4 in the second

Hence, in Case I: 8 students are divided into groups of 5 and 3 in $8 C_{3}$ ways.

Similarly, in Case II: 8 students are divided into two groups of 4 and 4 in $8 C_{4}$ ways.

Therefore, the total number of ways in which 8 students can travel is

$8 C_{3} + 8 C_{4}$ = 56 + 70 = 126.

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Q:

How many ways can 10 letters be posted in 5 post boxes, if each of the post boxes can take more than 10 letters ?

A) 5^10	B) 10^5
C) 5P5	D) 5C5

Answer & Explanation Answer: A) 5^10

Explanation:

Each of the 10 letters can be posted in any of the 5 boxes.

So, the first letter has 5 options, so does the second letter and so on and so forth for all of the 10 letters.

i.e. 5*5*5*….*5 (upto 10 times) = 5 ^ 10.

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Q:

A certain marathon had 50 people running for first prize, second, and third prize.How many ways are there to correctly guess the first, second, and third place winners?

A) 2	B) 1
C) 4	D) 3

Answer & Explanation Answer: B) 1

Explanation:

There is 1 way to correctly guess who comes in first, second, and third. There is only one set of first, second and third place winners. You must correctly guess these three people, and there is only one way to do so.

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Q:

Determine the total number of five-card hands that can be drawn from a deck of 52 cards.

A) 2589860	B) 2598970
C) 2598960	D) 2430803

Answer & Explanation Answer: C) 2598960

Explanation:

When a hand of cards is dealt, the order of the cards does not matter. If you are dealt two kings, it does not matter if the two kings came with the first two cards or the last two cards. Thus cards are combinations. There are 52 cards in a deck and we want to know how many different ways we can put them in groups of five at a time when order does not matter. The combination formula is used.

C(52,5) = 2,598,960

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Q:

Find the number of ways to take 20 objects and arrange them in groups of 5 at a time where order does not matter.?

A) 57090	B) 15540
C) 15504	D) 23670

Answer & Explanation Answer: C) 15504

Explanation:

${}^{20}C_{5}$ = 15504

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Q:

In how many ways can a group of 5 men and 2 women be made out of a total of 7 men and 3 women?

A) 135	B) 63
C) 125	D) 64

Answer & Explanation Answer: B) 63

Explanation:

Required number of ways = $(7 C_{5} * 3 C_{2}) = (7 C_{2} * 3 C_{1}) = 63$

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Q:

From 5 consonants and 4 vowels, how many words can be formed using 3 consonants and 2 vowels ?

A) 7600	B) 7200
C) 6400	D) 3600

Answer & Explanation Answer: B) 7200

Explanation:

From 5 consonants, 3 consonants can be selected in $5 C_{3}$ ways.

From 4 vowels, 2 vowels can be selected in $4 C_{2}$ ways.

Now with every selection, number of ways of arranging 5 letters is $5 P_{5}$ ways.

Total number of words = $5 C_{3} * 4 C_{2} * 5 P_{5}$

= 10x 6 x 5 x 4 x 3 x 2 x 1= 7200

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