Question 1: If ⁿC₈ = ⁿC₂, find ⁿC₂

Answer 1: It is known that, ⁿC_a = ⁿC_b ⇒ a=b or n=a+b

Therefore, ⁿC₈ = ⁿC_b ⇒ n = 8+2=10

ⁿC₂ = ¹⁰C₂ =

Question 2: Determine n if

(i) ²ⁿC₃ : ⁿC₃ = 12 : 1        (ii) ²ⁿC₃ : ⁿC₃ = 11 : 1

Answer 2:

(i) 

 

⇒ 2n – 1 = 3(n – 2)

⇒ 2n – 1 = 3n – 6

⇒ 3n -2n = -1 + 6

⇒ n = 5

(ii)

⇒ 4(2n-1) = 11(n-2)

⇒ 8n-4 = 11n-22

⇒ 11n-8n = -4+22

⇒ 3n = 18

⇒ n = 6

Question 3: How many chords can be drawn through 21 points on a circle?

Answer 3: For drawing one chord on a circle, only 2 points are required.

To know the number of chords that can be drawn through the given 21 points on a circle, the number of combinations have to be counted.

Therefore, there will be as many chords as there are combinations of 21 points taken 2 at a time.

Thus, required number of chords = ²¹C₂ =

Question 4: In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Answer 4: A team of 3 boys and 3 girls is to be selected from 5 boys and 4 girls.

3 boys can be selected from 5 boys in ⁵C₃ ways.

3 girls can be selected from 4 girls in ⁴C₃ ways.

Therefore, by multiplication principle, number of ways in which a team of 3 boys and 3 girls can be selected =⁵C₃ x ⁴C₃ =

= 10 x 4 = 40

Question 5: Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour.

Answer 5: There are a total of 6 red balls, 5 white balls, and 5 blue balls.

9 balls have to be selected in such a way that each selection consists of 3 balls of each colour.

Here,

3 balls can be selected from 6 red balls in ⁶C₃ ways.

3 balls can be selected from 5 white balls in ⁵C₃ ways.

3 balls can be selected from 5 blue balls in ⁵C₃ ways.

Thus, by multiplication principle, required number of ways of selecting 9 balls

= ⁶C₃  x ⁵C₃ x ⁵C₃ =

= 20 x 10 x 10 = 2000

Question 6: Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.

Answer 6: In a deck of 52 cards, there are 4 aces. A combination of 5 cards have to be made in which there is exactly one ace.

Then, one ace can be selected in ⁴C₁ ways and the remaining 4 cards can be selected out of the 48 cards in ⁴⁸C₄ ways.

Thus, by multiplication principle, required number of 5 card combinations

= ⁴⁸C₄ x ⁴C₁ =

= 778320

Question 7: In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers?

Answer 7: Out of 17 players, 5 players are bowlers.

A cricket team of 11 players is to be selected in such a way that there are exactly 4 bowlers.

4 bowlers can be selected in ⁵C₄ ways and the remaining 7 players can be selected

out of the 12 players in ¹²C₇  ways.

Thus, by multiplication principle, required number of ways of selecting cricket team

= ⁵C₄ x ¹²C₇ =

= 3960

Question 8: A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected.

Answer 8: There are 5 black and 6 red balls in the bag.

2 black balls can be selected out of 5 black balls in ⁵C₂ ways and 3 red balls can be selected out of 6 red balls in ⁶C₃ ways.

Thus, by multiplication principle, required number of ways of selecting 2 black and 3 red balls

= ⁵C₂ x ⁶C₃ =

= 10 x 20 = 200

Question 9: In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student?

Answer 9: There are 9 courses available out of which, 2 specific courses are compulsory for every student.

Therefore, every student has to choose 3 courses out of the remaining 7 courses. This can be chosen in ⁷C₃ ways.

Thus, required number of ways of choosing the programme

= ⁷C₃  =

= 35