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

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

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

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.

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.

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.

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 ;

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?

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 = 

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

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