Question: The letter of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the word LABOUR?

Solution:
Here,
The order of each letter in the dictionary is ABLORU.

Now,
with A in the beginning, the remaining letters can be permuted = 5! ways.
= 120 ways

Similarly,
with B in the beginning, the remaining letters can be permuted = 5! ways.
= 120 ways

With L in the beginning,
the first word will be LABORU, the second will be LABOUR.

Hence, the rank of the word LABOUR = 5! + 5! + 2
= 120 + 120 + 2
= 242

Question: Fifteen distinct points are randomly placed on the circumference of a circle. At most how many triangles can be formed using these points?
​
​Solution:
​Given that,
​Number of distinct points = 15
​Maximum number of triangles = 15C3
​= 15!/3!(15 - 3)!
​= (15 × 14 × 13 × 12!)/(3 × 2 × 12!)
= 455

Question: Two pipes A and B can fill the tank in 24 and 36 minutes, respectively. Both the pipes are opened together. After how many minutes should the pipe B be turned off, so that the tank be fill in 18 minutes?

Solution:

Given that,

Pipe A fills the tank in 24 minutes.

Pipe B fills the tank in 36 minutes.

Total time to fill the tank = 18 minutes.

Now,

LCM of 24 and 36 = 72 (Total capacity of the tank).

Efficiency of pipe A = 72/24 = 3 units/minute.

Efficiency of pipe B = 72/36 = 2 units/minute.

Let,

pipe B be turned off after x minutes.

Pipe A works for 18 minutes.

Pipe B works for x minutes.

Work done by A in 18 minutes = 3 × 18 = 54 units.

​Work done by B in x minutes = 2x = 2x units.

Total work done = 54 + 2x = 72

⇒ 2x = 72 - 54

⇒ 2x = 18

⇒ x = 18/2

∴ x = 9

∴ Pipe B should be turned off after 9 minutes.