Permutations & Combinations Questions and Answers

Q: In how many ways can three consonants and two vowels be selected from the letters of the word "TRIANGLE"?

TRIANGLE has consonants T, R, N, G, L and vowels I, A, E.\[ {}^{5}C_{3} \times {}^{3}C_{2} = 10 \times 3 = 30 \]Only the selection is asked for, not the arrangement, so combinations apply; arranging the five chosen letters would multiply this by \( 5! \).

Q: In a class there are 20 boys and 25 girls. In how many ways can a boy and a girl be selected?

The two choices are independent.\[ 20 \times 25 = 500 \]Every one of the 20 boys can be paired with every one of the 25 girls.

Q: The number of sequences in which 7 players can throw a ball, so that the youngest player may not be the last is -.

Count all sequences and remove the forbidden ones.\[ \text{total} = 7! = 5040 \]\[ \text{youngest last} = 6! = 720 \]\[ 5040 - 720 = 4320 \]Equivalently, the last place has 6 acceptable players and the rest arrange freely: \( 6 \times 6! = 4320 \).

Q: The number of new words that can be formed by rearranging the letters of the word 'ALIVE' is -.

ALIVE has five distinct letters.\[ 5! = 120 \text{ arrangements in all} \]One of those is the original word itself, which is not new:\[ 120 - 1 = 119 \]The 120 on offer is the total count and 24 is \( 4! \) - both a step away from what is asked.

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61.

A letter lock consists of three rings each marked with six different letters. The number of distinct unsuccessful attempts to open the lock is at the most -.

A.216
B.243
C.215
D.729

Answer: C

Each of the three rings has six letters, and they turn independently.

\[ 6 \times 6 \times 6 = 216 \text{ total settings} \]

Exactly one setting opens the lock, so the unsuccessful attempts number at most

\[ 216 - 1 = 215 \]

The "at most" matters: in the worst case every wrong setting is tried before the right one.

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62.

How many four digit numbers can be formed using the digits {1, 3, 4, 5, 7,9}(repetition of digits is not allowed)?

A.360
B.60
C.300
D.180

Answer: A

Six digits are available and four positions are to be filled without repetition.

\[ {}^{6}P_{4} = 6 \times 5 \times 4 \times 3 = 360 \]

None of the digits is 0, so there is no restriction on the leading position.

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63.

How many four digit even numbers can be formed using the digits {2, 3, 5, 1, 7, 9}

A.100
B.360
C.120
D.60

Answer: D

A number is even only if its last digit is even, and 2 is the only even digit available.

\[ \text{last digit} = 1 \text{ choice} \]

The other three positions take the remaining five digits without repetition:

\[ 5 \times 4 \times 3 = 60 \]

\[ 1 \times 60 = 60 \]

Fixing the constrained position first is the standard tactic for these.

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64.

In how many ways can three consonants and two vowels be selected from the letters of the word "TRIANGLE"?

A.25
B.13
C.40
D.30

Answer: D

TRIANGLE has consonants T, R, N, G, L and vowels I, A, E.

\[ {}^{5}C_{3} \times {}^{3}C_{2} = 10 \times 3 = 30 \]

Only the selection is asked for, not the arrangement, so combinations apply; arranging the five chosen letters would multiply this by \( 5! \).

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65.

In a class there are 20 boys and 25 girls. In how many ways can a boy and a girl be selected?

A.400
B.500
C.600
D.20

Answer: B

The two choices are independent.

\[ 20 \times 25 = 500 \]

Every one of the 20 boys can be paired with every one of the 25 girls.

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66.

A question paper consists of five problems, each problem having three internal choices. In how many ways can a candidate attempt one or more problems?

A.63
B.511
C.1023
D.15

Answer: C

For each problem the candidate has four options: attempt one of the three internal choices, or leave the problem alone.

\[ 4^{5} = 1024 \]

One of those is attempting nothing at all, which the question excludes:

\[ 1024 - 1 = 1023 \]

The 511 on offer comes from \( 2^{9} - 1 \) and ignores the internal choices.

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67.

In how many ways can five boys and three girls sit in a row such that all the boys sit together?

A.4800
B.5760
C.2880
D.15000

Answer: C

Treat the five boys as a single block, giving four units to arrange - the block and the three girls.

\[ 4! = 24 \]

Within the block the boys can be ordered in

\[ 5! = 120 \text{ ways} \]

\[ 24 \times 120 = 2880 \]

The block itself may sit anywhere among the girls, which is what the \( 4! \) counts.

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68.

The number of sequences in which 7 players can throw a ball, so that the youngest player may not be the last is -.

A.4000
B.2160
C.4320
D.5300

Answer: C

Count all sequences and remove the forbidden ones.

\[ \text{total} = 7! = 5040 \]

\[ \text{youngest last} = 6! = 720 \]

\[ 5040 - 720 = 4320 \]

Equivalently, the last place has 6 acceptable players and the rest arrange freely: \( 6 \times 6! = 4320 \).

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69.

A delegation of 5 members has to be formed from 3 ladies and 5 gentlemen. In how many ways the delegation can be formed, if 2 particular ladies are always included in the delegation?

A.20
B.54
C.42
D.60

Answer: A

The two particular ladies are already in, so only three more places remain and they come from the other six people (1 lady and 5 gentlemen).

\[ {}^{6}C_{3} = 20 \]

Fixing members reduces both the size of the pool and the number still to be chosen.

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70.

The number of new words that can be formed by rearranging the letters of the word 'ALIVE' is -.

A.24
B.23
C.119
D.120

Answer: C

ALIVE has five distinct letters.

\[ 5! = 120 \text{ arrangements in all} \]

One of those is the original word itself, which is not new:

\[ 120 - 1 = 119 \]

The 120 on offer is the total count and 24 is \( 4! \) - both a step away from what is asked.

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