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 ?

Correct Answer: 243

Explanation:

Digits are 0,1,2,3,4,5,6,7,8,9. So no. of digits are 10

 

First all possible case => 9(0 excluded) x 10 x 10 = 900

 

Second no repetition allowed =>9 x 9 x 8 = 648

 

Third all digits are same => 9 (111,222,333,444,555,666,777,888,999)

 

Three digit numbers where two of the three digits are same = 900 - 648 - 9 = 243 ;

Correct Answer: 36

Explanation:

Given there are 9 flavours of ice creams.

 

Each child takes the combination of two flavours and no two children will have the same combination

 

This can be done by  ways i.e children.

 

Number of children= C 2 9  = 9 x 8 / 1x 2 = 36.

Correct Answer: 125

Explanation:

Methods for selecting 4 questions out of 5 in the first section = 5 x 4 x 3 x 2 x 1/4 x 3 x 2 x 1 = 5. Similarly for other 2 sections also i.e 5 and 5

So total methods = 5 x 5 x 5 = 125.

Correct Answer: 253

Explanation:

The first person shakes hands with 22 different people, the second person also shakes hands with 22 different people, but one of those handshakes was counted in the 22 for the first person, so the second person actually shakes hands with 21 new people. The third person, 20 people, and so on...
So,
22 + 21 + 20 + 19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
= n(n+1)/2 = 22 x 23 /2 = 11 x 23 = 253.

Correct Answer: 325

Explanation:

Required number of signals =  5 P 1 + 5 P 2 + 5 P 3 + 5 P 4 + 5 P 5

= 5 + 20 + 60 + 120 + 120 = 325

 

 

Correct Answer: 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 = 

$$\begin{gathered} 6 C 3 x 7 C 2 + 6 C 4 x 7 C 1 + 6 C 5 \\ = 6 x 5 x 4 3 x 2 x 1 x 7 x 6 2 x 1 + 6 x 5 x 4 x 3 4 x 3 x 2 x 1 x 7 + 6 x 5 x 4 x 3 x 2 5 x 4 x 3 x 2 x 1 \\ = 420 + 105 + 6 \\ = 531 . \end{gathered}$$

Correct Answer: 164

Explanation:

For a triangle, we need 3 non-collinear points. So with 12 points (when all the 12 are such that any three non-collinear is 12 C 3. But among them 8 points are collinear.

 

If all these 8 points are different we get  8 C 3 triangles as they are collinear.

 

In  12 C 3 triangles, we do not get  8 C 3 triangles

 

Therefore, The number of triangles we get = 12 C 3 - 8 C 3 = 164

Correct Answer: 17280

Explanation:

There are 4 books on fairy tales and they have to be put together. They can be arranged in 4! ways.

 

Similarly, there are 5 novels.They can be arranged in 5! ways.

 

And there are 3 plays.They can be arranged in 3! ways.

 

So, by the counting principle all of them together can be arranged in 4!´5!´3! ways = 17280

Correct Answer: 100

Explanation:

1 million distinct 3 digit initials are needed.

 

Let the number of required alphabets in the language be ?n?.

 

Therefore, using ?n? alphabets we can form n * n * n =  n 3 distinct 3 digit initials.

 

Note distinct initials is different from initials where the digits are different.

 

For instance, AAA and BBB are acceptable combinations in the case of distinct initials while they are not permitted when the digits of the initials need to be different.

 

This  n 3 different initials = 1 million 

i.e.  n 3 = 10 6  (1 million = 10 6)

  => n =  10 2 = 100

 

Hence, the language needs to have a minimum of 100 alphabets to achieve the objective.

Correct Answer: 14

Explanation:

7 C 2- 7 = 14