Chapter 7 Permutations and Combinations Ex 7.4 Solutions
Question - 1 : - If nC8 = nC2,find nC2.
Answer - 1 : - 
Question - 2 : - Determine n if
(i) 2nC3:nC3 = 12: 1
(ii) 2nC3: nC3 = 11:1
Answer - 2 : -
Simplifying andcomputing
⇒ 4 × (2n – 1) =12 × (n – 2)
⇒ 8n – 4 = 12n –24
⇒ 12n – 8n = 24 –4
⇒ 4n = 20
∴ n = 5
⇒ 11n – 8n = 22 –4
⇒ 3n = 18
∴ n = 6
Answer - 3 : -
Given 21 points on a circle
We know that we require two points on the circle to draw a chord
∴ Number of chords is are
⇒ 21C2=
∴ Total number of chords can be drawn are 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 - 4 : -
Given 5 boys and 4girls are in total
We can select 3 boysfrom 5 boys in 5C3 ways
Similarly, we canselect 3 boys from 54 girls in 4C3 ways
∴ Number of ways ateam of 3 boys and 3 girls can be selected is 5C3 × 4C3
⇒ 5C3 × 4C3 =
⇒ 5C3 × 4C3 = 10 ×4 = 40
∴ Number of ways a teamof 3 boys and 3 girls can be selected is 5C3 × 4C3 =40 ways
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 : -
Given 6 red balls, 5white balls and 5 blue balls
We can select 3 redballs from 6 red balls in 6C3 ways
Similarly, we canselect 3 white balls from 5 white balls in 5C3 ways
Similarly, we canselect 3 blue balls from 5 blue balls in 5C3 ways
∴ Number of waysof selecting 9 balls is 6C3 ×5C3 × 5C3
∴ Number of waysof selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if eachselection consists of 3 balls of each colour is 6C3 ×5C3 × 5C3 =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 : -
Given a deck of 52cards
There are 4 Ace cardsin a deck of 52 cards.
According to question,we need to select 1 Ace card out the 4 Ace cards
∴ Number of waysto select 1 Ace from 4 Ace cards is 4C1
⇒ More 4 cards areto be selected now from 48 cards (52 cards – 4 Ace cards)
∴ Number of waysto select 4 cards from 48 cards is 48C4
∴ Number of 5 cardcombinations out of a deck of 52 cards if there is exactly one ace in eachcombination 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 : -
Given 17 players inwhich only 5 players can bowl if each cricket team of 11 must include exactly 4bowlers
There are 5 playershow bowl, and we can require 4 bowlers in a team of 11
∴ Number of waysin which bowlers can be selected are: 5C4
Now other players leftare = 17 – 5(bowlers) = 12
Since we need 11players in a team and already 4 bowlers are selected, we need to select 7 moreplayers from 12.
∴ Number of wayswe can select these players are: 12C7
∴ Total number ofcombinations possible are: 5C4 × 12C7
∴ Number of wayswe can select a team of 11 players where 4 players are bowlers from 17 playersare 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 : -
Given a bag contains 5black and 6 red balls
Number of ways we canselect 2 black balls from 5 black balls are 5C2
Number of ways we canselect 3 red balls from 6 red balls are 6C3
Number of ways 2 blackand 3 red balls can be selected are 5C2× 6C3
∴ Number of waysin which 2 black and 3 red balls can be selected from 5 black and 6 red ballsare 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 : -
Given 9 courses areavailable and 2 specific courses are compulsory for every student
Here 2 courses arecompulsory out of 9 courses, so a student need to select 5 – 2 = 3 courses
∴ Number of waysin which 3 ways can be selected from 9 – 2(compulsory courses) = 7 are 7C3
∴ Number of ways astudent selects 5 courses from 9 courses where 2 specific courses arecompulsory are: 35