A question paper consists of three sections 4,5 and 6 questions respectively. Attempting one question from each section is compulsory but a candidate need not attempt all the questions. In how many ways can a candidate attempt the questions?
At least 1 question from each section is compulsory, so from the 1st section the candidate can attempt 1 or 2 or 3 or 4 questions.
In each section each question can be dealt with in 2 ways, i.e. either he attempts it or leaves it.
So far 4 question there are 2 × 2 × 2 × 2 ways to attempt.
As he has to attempt at least 1 question, the total number of ways in which he can attempt questions from 1st section is 24 - 1
Similarly for the 2nd section there are 25 - 1 ways in which he can attempt and for the 3rd section there are 26 - 1 ways.
The ways in which the attempts one or more questions in any section is independent of the number of ways in which he attempts one or more questions from the other sections.
Thus, total number of ways in which he can attempt questions in that paper:
= (24 - 1)(25 - 1)(26 - 1)
= 15 × 31 × 63
In how many ways a President, VP and Water-boy can be selected from a group of 10 people.
Let there were x teams participating in the games, then total number of matches,
= nC2 = 153
On solving we get, n = −17 and n = 18
It cannot be negative so n = 18
A box contains 10 balls out of which 3 are red and rest are blue. In how many ways can a random sample of 6 balls be drawn from the bag so that at the most 2 red balls are included in the sample and no sample has all the 6 balls of the same colour?
Six balls can be selected in the following ways,
One red ball and 5 blue balls
Two red balls and 4 blue balls.
Total number of ways,
= (3C1 × 7C5) + (3C2 × 4C7)
= 63 + 105
Out of eight crew members three particular members can sit only on the left side. Another two particular members can sit only on the right side. Find the number of ways in which the crew can be arranged so that four men can sit on each side.