Answer: Option C

Explanation:
ⁿp_n = n!
⁶p₆ = 6 × 5 × 4 × 3 × 2 × 1 = 720

Answer: Option A

Explanation:
Two men, three women and one child can be selected in ⁴C₂ x ⁶C₃ x ⁵C₁ ways
= (4 x 3)/(2 x 1) x (6 x 5 x 4)/(3 x 2) x 5 
= 600 ways.

Answer: Option A

Explanation:
We know that, the number of straight lines that can be formed by the 11 points in which 6 points are collinear and no other set of three points, except those that can be selected out of these 6 points are collinear. 
Hence, the required number of straight lines 
¹¹C₂ - ⁶C₂ - ⁵C₂ + 1 + 1 = 55 - 15 - 10 -+ 2 = 32

Answer: Option D

Explanation:
The total number of ways of forming the group of ten representatives is ²²C₁₀ . 
The total number of ways of forming the group that consists of no seniors is ¹⁰C₁₀ = 1 way
The required number of ways = ²²C₁₀  - 1

Answer: Option B

Explanation:
The number of ways of selecting three men, two women and three children is: 
= ⁴C₃ x ⁶C₂ x ⁵C₃  
= (4 x 3 x 2)/(3 x 2 x 1) x (6 x 5)/(2 x 1) x (5 x 4 x 3)/(3 x 2 x 1) 
= 4 x 15 x 10 
= 600 ways.

Answer: Option D

Explanation:
Here, five seniors out of 12 seniors can be selected in ¹²C₅ ways. Also, five juniors out of ten juniors can be selected ¹⁰C₅ ways. Hence the total number of different ways of selection
= ¹²C₅ x ¹⁰C₅ = ¹²C₇ x ¹⁰C₅
= ¹²C₅ = ¹²C₇

Answer: Option C

Explanation:
Treat all boys as one unit. Now there are four students and they can be arranged in 4! ways. Again five boys can be arranged among themselves in 5! ways. 
Required number of arrangements = 4! * 5! = 24 x 120 = 2880.

Answer: Option C

Explanation:
x Not younger_______ ↑
The last ball can be thrown by any of the remaining 6 players. The first 6 players can throw the ball in ⁶C₆ ways.
The required number of ways = 6(6!) = 4320

Answer: Option A

Explanation:
There are three ladies and five gentlemen and a committee of 5 members to be formed.
Number of ways such that two ladies are always included in the committee = ⁶C₃ = (6 x 5 x 4)/6 = 20.

Answer: Option C

Explanation:
Number of words which can be formed = 5! - 1 = 120 - 1 = 119.