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Q:

# How many arrangements can be made out of the letters of the word COMMITTEE, taken all at a time, such that the four vowels do not come together?

 A) 216 B) 45360 C) 1260 D) 43200

Answer:   D) 43200

Explanation:

There are total 9 letters in the word COMMITTEE in which there are 2M's, 2T's, 2E's.

$\inline \therefore$ The number of ways in which 9 letters can be arranged = $\inline \frac{9!}{2!\times 2!\times 2!}$ = 45360

There are 4 vowels O,I,E,E in the given word. If the four vowels always come together, taking them as one letter we have to arrange 5 + 1 = 6 letters which include 2Ms and 2Ts and this be done in $\inline \frac{6!}{2!\times 2!}$ = 180 ways.

In which of 180 ways, the 4 vowels O,I,E,E remaining together can be arranged in $\inline \frac{4!}{2!}$ = 12 ways.

$\inline \therefore$ The number of ways in which the four vowels always come together = 180 x 12 = 2160.

Hence, the required number of ways in which the four vowels do not come together = 45360 - 2160 = 43200

Q:

The number of sequences in which 4 players can sing a song, so that the youngest player may not be the last is ?

 A) 2580 B) 3687 C) 4320 D) 5460

Answer & Explanation Answer: C) 4320

Explanation:

Let 'Y' be the youngest player.

The last song can be sung by any of the remaining 3 players. The first 3 players can sing the song in (3!) ways.

The required number of ways = 3(3!) = 4320.

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Q:

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) 215 B) 268 C) 254 D) 216

Answer & Explanation Answer: A) 215

Explanation:

Since each ring consists of six different letters, the total number of attempts possible with the three rings is = 6 x 6 x 6 = 216. Of these attempts, one of them is a successful attempt.

Maximum number of unsuccessful attempts = 216 - 1 = 215.

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Q:

The number of ways in which six boys and six girls can be seated in a row for a photograph so that no two girls sit together is  ?

 A) 2(6!) B) 6! x 7 C) 6! x ⁷P₆ D) None

Answer & Explanation Answer: C) 6! x ⁷P₆

Explanation:

We can initially arrange the six boys in 6! ways.
Having done this, now three are seven places and six girls to be arranged. This can be done in ⁷P₆ ways.

Hence required number of ways = 6! x ⁷P₆

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Q:

The number of permutations of the letters of the word 'MESMERISE' is  ?

 A) 9!/(2!)^{2}x3! B) 9! x 2! x 3! C) 0 D) None

Answer & Explanation Answer: A) 9!/(2!)^{2}x3!

Explanation:

n items of which p are alike of one kind, q alike of the other, r alike of another kind and the remaining are distinct can be arranged in a row in n!/p!q!r! ways.
The letter pattern 'MESMERISE' consists of 10 letters of which there are 2M's, 3E's, 2S's and 1I and 1R.
Number of arrangements = $\inline&space;\fn_jvn&space;\small&space;\frac{9!}{(2!)^{2}x3!}$

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Q:

A group of 10 representatives is to be selected out of 12 seniors and 10 juniors. In how many different ways can the group be selected if it should have at least one senior ?

 A) ²²C₁₀ + 1 B) ²²C₉ + ¹⁰C₁ C) ²²C₁₀ D) ²²C₁₀ - 1

Answer & Explanation Answer: D) ²²C₁₀ - 1

Explanation:

The total number of ways of forming the group of ten representatives is ²²C₁₀.
The total number of ways of forming the group that consists of no seniors is ¹⁰C₁₀ = 1 way
The required number of ways = ²²C₁₀ - 1

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