Answer: 120960

Explanation:
In the word 'MATHEMATICS', we treat the vowels AEAI as one letter.
Thus, we have MTHMTCS (AEAI).
Now, we have to arrange 8 letters, out of which M occurs twice, T occurs twice and the rest are different.
Number of ways of arranging these letters = 8! / ((2!)(2!)) = 10080.

Now, AEAI has 4 letters in which A occurs 2 times and the rest are different.
Number of ways of arranging these letters = 4!/2! = 12.

Required number of words = (10080 x 12) = 120960.

Answer: 36

Explanation:
There are 6 letters in the given word, out of which there are 3 vowels and 3 consonants.
Let us mark these positions as under:
(1) (2) (3) (4) (5) (6)
Now, 3 vowels can be placed at any of the three places out 4, marked 1, 3, 5.
Number of ways of arranging the vowels = 3P3 = 3! = 6.
Also, the 3 consonants can be arranged at the remaining 3 positions.
Number of ways of these arrangements = 3P3 = 3! = 6.
Total number of ways = (6 x 6) = 36.

Answer: 209

Explanation:
We may have (1 boy and 3 girls) or (2 boys and 2 girls) or (3 boys and 1 girl) or (4 boys).
Required number
of ways = (6C1 x 4C3) + (6C2 x 4C2) + (6C3 x 4C1) + (6C4)
= (6C1 x 4C1) + (6C2 x 4C2) + (6C3 x 4C1) + (6C2)
= (6 x 4) + 6 x 5 x 4 x 3 + 6 x 5 x 4 x 4 + 6 x 5
2 x 1 2 x 1 3 x 2 x 1 2 x 1

= (24 + 90 + 80 + 15)
= 209.

Answer: 756

Explanation:

Answer: 10584

Explanation:
4 boys out of 9 can be selected in 9C4 ways = 126

3 girls out of 9 can be selected in 9C3 ways = 84

Required number of ways = 126*84 = 10584

Answer: 875

Explanation:
The minimum requirement is of 1 buffalo. Hence, 1 or more than 1 buffalo can be selected with no restriction on the number of cows. The possible combinations are:
(1 buffalo, 3 cows), (2 buffaloes, 2 cows), (3 buffaloes, 1 cow), (4 buffaloes, 0 cows).
Required number of ways of selection = (5C1*9C3) + (5C2*9C2) + (5C3*9C1) + (5C4)

= 420 + 360 + 90 + 5
= 875

Answer: 210

Explanation:
The total number of members is 4 + 6 or 10. A committee of six
=>can be formed from 10 in 10C6 ways = (10C4) or 210 ways

Answer: 1296

Explanation:
the number of rectangles that can be formed on an
=> n × n chessboard (R) = [n (n+1) / 2]2
=> As n = 8, R = 362 = 1296

Answer: 190

Explanation:
The number of lines that can be drawn, joining n points on a plane
 nC2 (when no three points are collinear). Here n = 20
required number of lines = 20C2 = 190

Answer: 9! /2 

Explanation:
n beads can be strung in a necklace in (n – 1)! / 2 ways
=> Here n = 10
Required number of ways = 9! /2