A Cricket team of 23 people all shake hands with each other exactly once. How many hand shakes occur ?

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: 253

Explanation:

Number of non - negative integral solutions = n + r - 1 C r - 1 = C 2 23 = 253 

Correct Answer: 1587600

Explanation:

 H   L   C   N    T    A   U   I   O

 

      L        N          A        I

 

There are total 131 letters out of which 7 are consonants and 6 are vowels. Also ther are 2L's , 2N's, 2A's and 2I's.

 

If all the consonants  are together then the numberof arrangements =  7 ! 2 !x 1/2! .

But the 7 consonants  can be arranged themselves in   7 ! 2 ! x 1/2! ways. 

 

Hence the required number of ways =  7 ! 2 ! 2 ! 2 = 1587600

Correct Answer: 21600

Explanation:

4 novels can be selected out of 5 in  5 C 4 ways.

2 biographies can be selected out of 4 in  4 C 2 ways.

Number of ways of arranging novels and biographies = 5 C 4 * 4 C 2  = 30

After selecting any 6 books (4 novels and 2 biographies) in one of the 30 ways, they can be arranged on the shelf in 6! = 720 ways.

By the Counting Principle, the total number of arrangements = 30 x 720 = 21600

Correct Answer: 720

Explanation:

There are 7 letters in the word ?Bengali of these 3 are vowels and 4 consonants.

 

Considering vowels a, e, i as one letter, we can arrange 4+1 letters in 5! ways in each of which vowels are together. These 3 vowels can be arranged among themselves in 3! ways.

 

Total number of words = 5! x 3!= 120 x 6 = 720

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: 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: 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: 6! x ?P?

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 ?P? ways.

Hence required number of ways = 6! x ?P?

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 = 

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