Rajasthan Board RBSE Class 11 Maths Chapter 6 Permutations and Combinations Miscellaneous Exercise
If nPn-2 = 60, then n will be:
nPr ÷ nCr is equal to:
(B) (n – r)!
Hence, the option (D) is correct.
How many ways 5 persons can sit around the round table:
Total number of ways to sitting 5 persons around the round table
= (5 – 1)! = 4!
= 4 × 3 × 2 × 1 = 24.
Hence, option (B) is correct.
How many words can be formed using letters BHILWARA:
There are 8 letters in the given word in which 2 A’s and other letters are different.
Then numbers of words formed by letters of BHILWARA =
Hence option (A) is correct.
Find the value of 61C57 – 60C56:
If 15C3r = 15Cr+3, then r is equal to:
There are 6 points on the circumference of a circle, the number of straight lines joining their points will be:
Number of lines passing through n points = nC2
Number of lines passing through 6 points
Hence, option (B) is correct.
How many words can be formed using letters of BHOPAL?
Here, the number of letters are 6 and every time we take 6 letters.
Hence, required number of = ⌊6 = 6 × 5 × 4 × 3 × 2 × 1 = 720
Hence, option (D) is correct.
There are 4 points on the circumference of a circle by joining them how many triangles can be formed?
There are three vertices in a triangle.
Given: There are 4 points on the circumference of a circle.
Then, the number of required triangles
Hence, option (A) is correct.
If nC2 = nC7, then find nC16
Find the value of n:
(i) nC2 : nC2 = 12 : 1
(ii) 2nC3 : nC3 = 11 : 1
Find the number of chords passing through 11 points on the circumference of a circle.
Number of the point on the circumference of circle = 11
We know that a chord is formed by joining two points.
Hence taking 2 points out of 11 points number of chords = 11C2
Hence, the number of chords = 55.
Determine the number of combinations out of deck of 52 cards of each selection of 5 cards has exactly one ace.
Number of cards in a deck = 52
Total number of aces = 4
Hence Number of remaining cards = 52 – 4 = 48
We have to make a collection of 5 cards, in which there is 1 ace and 4 other cards.
Hence, number of ways choosing 1 out of 4 ace.
Now,Number of ways choosing 4 cards out of 48 cards
Total number of collection having 5 cards
= 4C1 × 48C4 = 4 × 194580 = 778320
Hence, number of combinations formed by 5 cards = 778320.
There are n points in a plane in which m points are collinear. How many triangles will be formed by joining three points?
For making a triangle we need three points thus if 3 points out of n points are not in a line, then nC3 triangle can be formed by n points but m point is in a line, thus mC3 triangle forms less.
Hence, required number of triangles = nC3 – mC3.
Find the number of diagonals of a decagon.
There are 10 vertices in a decagon. Now by joining two vertices, we get a side of decagon or a diagonal.
Hence, the number of the line segment by joining of vertices of decagon = number of ways taking 2 points out of 10 points
Hence, the number of diagonals is 35.
There are 5 empty seats in a train, then in how many ways three travellers can sit on these seats?
A number of ways of seating of 3 travellers out of 5 seats.
Hence, 3 travellers can be seated in 60 ways.
A group of 7 has to be formed from 6 boys and 4 girls. In how many ways a group can be formed if boys are in the majority in this group?
Number of boys = 6
Number of girls = 4
Number of members in the group = 7
One putting boys in the majority in the group
We have to choose in the following ways :
4 boys + 3 girls
5 boys + 2 girls
6 boys + 1 girls
Number of ways choosing 4 boys and 3 girls = 6C4 × 4C3
Number of ways choosing 5 boys and 2 girls = 6C5 × 4C2
Number of ways choosing 6 boys and 1 girl = 6C6 × 4C1
Required number of ways forming the group
Hence, the group can be formed in 100 ways.
In the conference of 8 persons, every person handshake with each other only once, then find the total number of hand shook.
When two persons shake hand with each other, then this is taken as one handshake.
Hence, the number of handshakes is equal to choosing 2 persons out of 8 persons.
Number of handshake
Hence, the required number of hand shaken is 28.
In how many ways 6 men and 6 women can sit around the round table whereas any two women never sit together?
First, we sit the men in an order that there is a place vacant between two men.
Then, number of ways sitting 6 men = (6 – 1)! = 5!
Now a number of ways sitting 6 women in the vacant places = 6!
Number of permutations = (6 – 1)! = 5!
Now, the number of ways sitting 6 women in the vacant place = 6!
Number of permutaions = 5! × 6!
= (5 × 4 × 3 × 2 × 1) × (6 × 5 × 4 × 3 × 2 × 1)
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S are together?
Number of letters in word ASSASSINATION = 13
In this word A comes 3 times, S comes 4 times, I comes 2 times, N comes 2 times and other letters are different.
Now, taking 4’s together, then consider it as one letter, now adding other 9 letters and thus the permutation formed by 10 letters
[Because I comes 2 times, N comes 2 times and A comes 3 times]
Hence, the total number of ways is 151200.