**Exercise 6.4**

Permutations and combinations

**Question and Answers**

**Class 11 – Maths**

Permutations and combinations

Class | Class 11 |

Subject | Mathematics |

Chapter Name | Permutations and Combinations |

Chapter No. | Chapter 6 |

Exercise | Exercise 6.4 |

Category | Class 11 Maths NCERT Solutions |

**Question 1 If ^{n}C_{8} = ^{n}C_{2}, find ^{n}C_{2}.**

**Answer** ^{n}C_{a} = ^{n}C_{b}

⇒ a = b or n = a+ b

n = 8 + 2 =10

Therefore

**Question 2 Determine n , if**

**(i) ^{2n}C_{3} = ^{n}C_{3} = 12 : 1 **

⇒ 2n – 1 = 3(n – 2)

⇒ 2n – 1 = 3n – 6

⇒ 3n – 2n = -1 + 6

⇒ n = 5

**(ii) ^{2n}C_{3} = ^{n}C_{3} = 11 : 1 **

⇒ 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 **For drawing one chord 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, the chords can be drawn through 21 points taken 2 as equal to each chord.

Thus, required number of chords =

⇒ 210

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

**Answer ** 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 ^{5}C_{3} ways.

3 girls can be selected from 4 girls in ^{4}C_{3} ways.

Number of ways in which a team of 3 boys and 3 girls can be selected =

⇒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 **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 ^{6}C_{3} ways.

3 balls can be selected from 5 white balls in ^{5}C_{3} ways.

3 balls can be selected from 5 blue balls in ^{5}C_{3} ways.

Required number of ways of selecting the 9 balls will be –

∴ 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 is ^{6}C_{3} ×^{5}C_{3} × ^{5}C_{3} = 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** We have a deck of 52 cards, which contains 4 aces. When we make a combination, 5 cards should be made in such a manner that there is exactly one ace.

Then, one ace can be selected in ^{4}C_{1} ways and the remaining 4 cards can be selected out of the 48 cards in ^{48}C_{4} ways.

The required number of 5 card combinations will be –

**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 **The number of players out of which we have to select is 17 players and only 5 players from them 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 ^{5}C_{4} ways and the remaining 7 players can be selected out of the 12 players in ^{12}C_{7} ways.

Required number of ways of selecting cricket team –

**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 **There are 5 black and 6 red balls in the bag.

2 black balls can be selected out of 5 black balls in ^{5}C_{2} ways and 3 red balls can be selected out of 6 red balls in ^{6}C_{3} ways.

Required number of ways of selecting 2 black and 3 red balls –

**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 **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 ^{7}C_{3} ways.

Thus, required number of ways of choosing the programme –

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