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Counting Techniques: ReviewKevin Carl P. Santos

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Question # 1

• A child psychologist prepares four-letter words to be used in a memory test. How many words can she construct if she does not repeat letters and limits her choices to the letters a, b, d, e, i, l, o, and p? How many of these words begin with the letter a? How many have vowels as the second and fourth letters?

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Question # 2

• How many four-letter code words can be formed from the letters of the English alphabet if consonants and vowels are to alternate and repetitions are not allowed?

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Question # 3

• How many five-digit numbers can be formed from the digits 1, 2, 3, 4, 5, and 6 if no repetitions are allowed?

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Question # 4

• A class consists of 10 men and 20 women. An examination is given, and the students are ranked according to their performance. Assume that no two students obtain the same score.

a) How many different rankings are possible?b) If the men were ranked just among themselves

and the women among themselves, how many different rankings are possible?

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Question # 5

• How many different letter arrangements can be made from the letters of the word MISSISSIPPI?

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Question # 6

• How many ways can a man divide 7 gifts among his 3 children if the eldest is to receive 3 gifts and the others 2 each?

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Question # 7

• How many ways can you order the letters of TORONTO if it begins with exactly two O’s?

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Question # 8• How many ways can 8 people be seated in a row

if

a) there are no restrictions on the seating arrangement;

b) persons A and B must sit next to each other;c) there are 4 men and 4 women and no 2 men or

2 women can sit next to each other;d) there are 5 men and they must sit next to each

other;

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Question # 9Suppose a class consists of 30 students, 10 of which are

men and the remaining 20 are women. Suppose a sample of 10 students will be selected, how many possible

• ordered samples with replacement are there?• ordered samples without replacement are there?• unordered samples without replacement are there?• ordered samples with replacement are there where 3 of

the students selected are men?• ordered samples without replacement are there where 3

of the students selected are men?• ordered samples without replacement where at most 3 of

the students selected are men?

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Question # 10

• How many ways can you order the letters in QUEST if the vowels must never be together?

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Question # 11• Suppose there are 20 red apples, 15 green apples and 10

yellow apples. Let us assume that apples of the same color cannot be distinguished from each other.

a. How many ways can you arrange the 45 apples in a row?b. How many ways can you distribute the 20 red apples to

5 children if each child must receive at least 1 apple?c. How many ways can you distribute all the red apples,

green apples and yellow apples to 5 children if each child must receive at least 1 apple of each color?

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Question # 12• A box contains 25 numbered balls, of which 15 are

red and 10 are white. A sample of 5 balls is to be selected without replacement.

a. How many different ordered samples are possible?b. How many samples in (a) contain all red balls?c. How many samples in (a) contain 3 red balls and 2

white balls?d. How many samples in (a) contain at least 4 red

balls?

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Question # 13

From a group of five men and three women, how many ways can you form a committee consisting of 3 people

• with no restrictions?• with 2 men and 1 woman?

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Question # 14

• Ten identical apples are to be distributed among 4 children, how many divisions are possible? How many, if each child must receive at least one apple.

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Question # 15

• How many distinct nonnegative integer-valued solution of x1+x2+x3=10 are possible?

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Question # 16

• If 8 identical blackboards are to be divided among 4 schools, how many divisions are possible? How many, if each school must receive at least 1 blackboard?

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Question # 17

• An elevator starts at the basement with 8 people (not including the elevator operator) and discharges them all by the time it reaches the top floor, number 6. In how many ways could the operator have perceived the people leaving the elevator if all people look alike to him? What if the 8 people consisted of 5 men and 3 women and the operator could tell a man from a woman?

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Question # 18• Four red balls are marked R1, R2, R3, R4. Five blue

balls are marked B1, B2, B3, B4, B5. Three white balls are marked W1, W2, W3.

a) How many combinations of size 5 can be formed?

b) Among these combinations of size 5, how many contain 3 red balls?

c) Among these combinations of size 5, how many contain 2 red balls, 2 blue balls, and 1 white ball?

d) Among these combinations of size 5, how many do not contain any red ball?

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Question # 19

• There are 5 blue boxes, 3 green boxes, and 2 red boxes. Determine the number of ways that these boxes can be arranged in a row under each of the following conditions:▫ all the boxes are distinct▫ the blue boxes are identical, green boxes are

identical and the red boxes are identical

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Question # 20• A poker hand consists of 5 cards from a standard

deck of 52 cards.▫ How many possible poker hands are there if we

ignore the order in which the cards are dealt?▫ How many among the poker hands in (a) consist

of exactly 2 hearts?▫ How many among the poker hands in (a) consist

of 2 hearts, 2 spades and 1 club?▫ How many among the poker hands in (a) consist

of 5 cards that are all of the same suit?

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Question # 21• In the PCSO Lotto 6/42, a 6-number combination is

drawn from a lot of numbers from 1 to 42 at random. To win a prize, at least three of the player’s chosen numbers must match with those of the six winning numbers.▫ How many 6-number combinations are possible in

this lotto?▫ How many 6-number combinations in (a) contain

exactly 3 of the winning numbers?▫ How many 6-number combinations in (a) contain

at least 3 of the winning numbers?

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