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.
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.
Required number of signals =
= 5 + 20 + 60 + 120 + 120 = 325
Number of non - negative integral solutions = = 253
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 = x 1/2! .
But the 7 consonants can be arranged themselves in x 1/2! ways.
Hence the required number of ways = = 1587600
4 novels can be selected out of 5 in
ways.
2 biographies can be selected out of 4 in
ways.
Number of ways of arranging novels and biographies =
= 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
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= = 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 =
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 . But among them 8 points are collinear.
If all these 8 points are different we get triangles as they are collinear.
In triangles, we do not get triangles
Therefore, The number of triangles we get = = 164
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