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Q:

How many number of times will the digit ‘7' be written when listing the integers from 1 to 1000?

A) 243 B) 300
C) 301 D) 290

Answer:   B) 300



Explanation:

7 does not occur in 1000. So we have to count the number of times it appears between 1 and 999. Any number between 1 and 999 can be expressed in the form of xyz where 0 < x, y, z < 9.

 

1. The numbers in which 7 occurs only once. e.g 7, 17, 78, 217, 743 etc

 

This means that 7 is one of the digits and the remaining two digits will be any of the other 9 digits (i.e 0 to 9 with the exception of 7)

 

You have 1*9*9 = 81 such numbers. However, 7 could appear as the first or the second or the third digit. Therefore, there will be 3*81 = 243 numbers (1-digit, 2-digits and 3- digits) in which 7 will appear only once.

 

In each of these numbers, 7 is written once. Therefore, 243 times.

 

 

2. The numbers in which 7 will appear twice. e.g 772 or 377 or 747 or 77

 

In these numbers, one of the digits is not 7 and it can be any of the 9 digits ( 0 to 9 with the exception of 7).

 

There will be 9 such numbers. However, this digit which is not 7 can appear in the first or second or the third place. So there are 3 * 9 = 27 such numbers.

 

In each of these 27 numbers, the digit 7 is written twice. Therefore, 7 is written 54 times.

 

 

3. The number in which 7 appears thrice - 777 - 1 number. 7 is written thrice in it.

 

Therefore, the total number of times the digit 7 is written between 1 and 999 is

 

243 + 54 + 3 = 300

Q:

From a group of 7 boys and 6 girls, five persons are to be selected to form a team, so that at least 3 girls are there in the team. In how many ways can it be done?

A) 427 B) 531
C) 651 D) 714
 
Answer & Explanation Answer: B) 531

Explanation:

Given in the question that, there are 7 boys and 6 girls. 

Team members = 5

Now, required number of ways in which a team of 5 having atleast 3 girls in the team = 

6C3  x 7C2  + 6C4 x 7C1 + 6C5= 6x5x43x2x1 x 7x62x1 + 6x5x4x34x3x2x1 x 7 + 6x5x4x3x25x4x3x2x1= 420 + 105 + 6= 531.

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2 138
Q:

The number of ways in which 8 distinct toys can be distributed among 5 children?

A) 5P8 B) 5^8
C) 8P5 D) 8^5
 
Answer & Explanation Answer: B) 5^8

Explanation:

As the toys are distinct and not identical,

For each of the 8 toys, we have three choices as to which child will receive the toy. Therefore, there are 58 ways to distribute the toys.

 

Hence, it is 58 and not 85.

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1 650
Q:

In how many different ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together?

A) 1440 B) 720
C) 2250 D) 3600
 
Answer & Explanation Answer: D) 3600

Explanation:

Given word is THERAPY.

Number of letters in the given word = 7

These 7 letters can be arranged in 7! ways.

Number of vowels in the given word = 2 (E, A)

The number of ways of arrangement in which vowels come together is 6! x 2! ways

 

Hence, the required number of ways can the letters of the word 'THERAPY' be arranged so that the vowels never come together = 7! - (6! x 2!) ways = 5040 - 1440 = 3600 ways.

 

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1 368
Q:

In how many different ways can the letters of the word 'HAPPYHOLI' be arranged?

A) 89,972 B) 90,720
C) 72,000 D) 81,000
 
Answer & Explanation Answer: B) 90,720

Explanation:

The given word HAPPYHOLI has 9 letters

These 9 letters can e arranged in 9! ways.

But here in the given word letters H & P are repeated twice each

Therefore, Number of ways these 9 letters can be arranged is 

9!2! x 2! = 9 x 8 x 7 x 6 x 5 x 4 x 32 = 90,720 ways.

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5 531
Q:

How many words can be formed with or without meaning by using three letters out of k, l, m, n, o without repetition of alphabets.

A) 60 B) 120
C) 240 D) 30
 
Answer & Explanation Answer: A) 60

Explanation:

Given letters are k, l, m, n, o = 5

number of letters to be in the words = 3

Total number of words that can be formed from these 5 letters taken 3 at a time without repetation of letters = 5P3 ways.

 5P3 = 5 x 4 x 3 = 60 words.

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7 519
Q:

The letters of the word PROMISE are to be arranged so that three vowels should not come together. Find the number of ways of arrangements?

A) 4320 B) 4694
C) 4957 D) 4871
 
Answer & Explanation Answer: A) 4320

Explanation:

Given Word is PROMISE.

Number of letters in the word PROMISE = 7

Number of ways 7 letters can be arranged = 7! ways

Number of Vowels in word PROMISE = 3 (O, I, E)

Number of ways the vowels can be arranged that 3 Vowels come together = 5! x 3! ways

 

Now, the number of ways of arrangements so that three vowels should not come together

= 7! - (5! x 3!) ways = 5040 - 720 = 4320.

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7 978
Q:

In how many different ways can the letters of the word 'POVERTY' be arranged ?

A) 2520 B) 5040
C) 1260 D) None
 
Answer & Explanation Answer: B) 5040

Explanation:

The 7 letters word 'POVERTY' be arranged in 7P7 ways = 7! = 5040 ways.

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12 575
Q:

A decision committee of 5 members is to be formed out of 4 Actors, 3 Directors and 2 Producers. In how many ways a committee of 2 Actors, 2 Directors and 1 Producer can be formed ?

A) 18 B) 24
C) 36 D) 32
 
Answer & Explanation Answer: C) 36

Explanation:

Required Number of ways = 4C2 × 3C2 × 2C1 = 36 = 36

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17 648