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Mathmetics

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Q11.	There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. In how many ways can the committee be formed ?

1)	8750	2)	10420
3)	11760	4)	12420
5)	None of these

Answer : 11760
Explanation :

We need to select 5 men from 8 men and 6 women from 10 women

Number of ways to do this = ⁸C₅ x ¹⁰C₆

= ⁸C₃ x ¹⁰C₄ [Applied the formula nCr = nC(n - r) ]

= (8�7�63�2�1) (10�9�8�74�3�2�1)

= 56 x 210 = 11760

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Q12.	Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed ?

1)	2100	2)	24400
3)	21300	4)	25200
5)	None of these

Answer : 25200
Explanation :

Number of ways of selecting 3 consonants out of 7 = ⁷C₃

Number of ways of selecting 2 vowels out of 4 = ⁴C₂

Number of ways of selecting 3 consonants out of 7 and 2

vowels out of 4 = ⁷C₃ x ⁴C₂

=(7�6�5/3�2�1)�(4�3/2�1)=210

It means that we can have 210 groups where each group

contains total 5 letters(3 consonants and 2 vowels).

Number of ways of arranging 5 letters among themselves = 5! = 5 x 4 x 3 x 2 x 1 = 120

Hence, Required number of ways = 210 x 120 = 25200

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Q13.	In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together ?

1)	100800	2)	120960
3)	240150	4)	4989600
5)	None of thesae

Answer : 120960
Explanation :

In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.

Thus, we have MTHMTCS (AEAI).

Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.

Number of ways of arranging these letters = 8! / (2!)(2!)= 10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.

Number of ways of arranging these letters = 4! /2! = 12.

So, required number of words = (10080 x 12) = 120960.

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Q14.	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 ?

1)	42	2)	48
3)	63	4)	90
5)	None of these

Answer : 63
Explanation : The required number of ways = (⁷C₅ x³C₂) = (⁷C₂ x ³C₁) = (7 x 6 )/(2 x 1) x 3 = 63.

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Q15.	In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions ?

1)	24	2)	32
3)	36	4)	48
5)	None of these

Answer : 36
Explanation :

There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.

Let us mark these positions as under:

(1) (2) (3) (4) (5) (6)

Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.

Number of ways of arranging the vowels =³P₃ = 3! = 6.

Also, the 3 consonants can be arranged at the remaining 3 positions.

Number of ways of these arrangements = ³P₃ = 3! = 6.

Total number of ways = (6 x 6) = 36.

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Q16.	How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated ?

1)	6	2)	10
3)	12	4)	20
5)	None of thse

Answer : 20
Explanation :

Since each desired number is divisible by 5, so we must have 5 at the unit place. So, there is 1 way of doing it.

The tens place can now be filled by any of the remaining 5 digits (2, 3, 6, 7, 9). So, there are 5 ways of filling the tens place.

The hundreds place can now be filled by any of the remaining 4 digits. So, there are 4 ways of filling it.

So, required number of numbers = (1 x 5 x 4) = 20.

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Q17.	In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together ?

1)	810	2)	1440
3)	2880	4)	50400
5)	57600

Answer : 50400
Explanation :

In the word 'CORPORATION', we consider the vowels OOAIO as one letter.

Thus, we have CRPRTN (OOAIO).

This has 7 (6 + 1) letters of which R occurs 2 times and the rest are different.

Number of ways arranging these letters =7!/2! = 2520.

Now, 5 vowels in which O occurs 3 times and the rest are different, can be arranged in 5!/3!= 20 ways.

So, required number of ways = (2520 x 20) = 50400.

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Q18.	In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together ?

1)	360	2)	480
3)	720	4)	1280
5)	None of these

Answer : 720
Explanation :

The word 'LEADING'' has 7 different letters.

When the vowels EAI are always together, they can be supposed to form one letter.

Then, we have to arrange the letters LNDG (EAI).

Now, 5 (4 + 1 = 5) letters can be arranged in 5! = 120 ways.

The vowels (EAI) can be arranged among themselves in 3! = 6 ways.

So, required number of ways = (120 x 6) = 720.

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Q19.	From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done ?

1)	565	2)	664
3)	742	4)	756
5)	None of these

Answer : 756
Explanation :

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Mathmetics

Q11.

There are 8 men and 10 women and you need to form a committee of 5 men and 6 women. In how many ways can the committee be formed ?

Q12.

Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed ?

Q13.

In how many different ways can the letters of the word 'MATHEMATICS' be arranged so that the vowels always come together ?

Q14.

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 ?

Q15.

In how many different ways can the letters of the word 'DETAIL' be arranged in such a way that the vowels occupy only the odd positions ?

Q16.

How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, 7 and 9, which are divisible by 5 and none of the digits is repeated ?

Q17.

In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together ?

Q18.

In how many different ways can the letters of the word 'LEADING' be arranged in such a way that the vowels always come together ?

Q19.

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done ?

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