Permutations and Combinations — Class 11 solved problems

49 problems from this chapter, each solved step by step.

Find the probability that in a random arrangement of the letters of the word MATHEMATICS, the two A's are not together.
The probability that the two A's are not together is (9)/(11).
36. Find the number of ways in which six people can ride a toboggan if one of a subset of three must drive. 37. (a) Find the number of ways in which five person
The number of ways in which six people can ride a toboggan if one of a subset of three must drive is 360.
In how many ways can 4 red, 3 yellow and 2 green discs be arranged in a row if the discs of the same colour are indistinguishable?
The discs can be arranged in 1260 ways.
How many numbers lying between 100 and 1000 can be formed with the digits 0, 1, 2, 3, 4, 5, if the repetition of the digits is not allowed?
100
How many ways can a 2×n rectangle be tiled using 1×2 dominoes?
The number of ways to tile a 2 × n rectangle using 1 × 2 dominoes is F_n+1, where F_n is the n -th Fibonacci number with F_1=1 and F_2=1.
The largest n N such that 3^n divides 50! is:
22
How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed?
3024
The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is
100
Find the number of different signals that can be generated by arranging at least 2 flags in order (one below the other) on a vertical staff, if five different f
320
Find the value of n such that i) ^nP_5 =42 ^nP_3, n>4 ii) (^nP_4)/(^(n-1)P_4), n>4
(i) n=10, (ii) (n)/(n-4)
Find the number of 4 letter words, with or without meaning, which can be formed out of the letters of the word ROSE, where the repetition of the letters is not
24
The number of different 5-digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last
4608
Line L_1 of slope 2 and line L_2 of slope 12 intersect at the origin O. In the first quadrant, P_1,P_2,,P_12 are 12 points on L_1 and Q_1,Q_2,,Q_9 are 9 points
1134
Find the number of permutations of the letters of the word ALLAHABAD.
The number of permutations of the letters of the word ALLAHABAD is 7560.
Number of functions f: 1, 2,, 100 → 0, 1, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
392
how many words combinations can you make with the name Rakesh Ramesh
The total number of word combinations that can be made with the name Rakesh Ramesh is 29,937,600.
A committee of 5 is to be formed from 6 men and 4 women with at least 3 women. How many ways?
The total number of ways to form the committee is 66.
Find the number of different 8-letter arrangements that can be made from the letters of the word DAUGHTER so that (i) all vowels occur together (ii) all vowels
(i) 4320, (ii) 36000
In how many ways can 5 girls and 3 boys be seated in a row so that no two boys are together?
14400
The number of words, which can be formed using all the letters of the word DAUGHTER, so that all the vowels never come together, is
36000
Find the number of ways to color the vertices of a regular hexagon with 3 colors such that no two adjacent vertices have the same color.
The number of ways to color the vertices of a regular hexagon with 3 colors such that no two adjacent vertices have the same color is 66.
Find the number of words with or without meaning which can be made using all the letters of the word AGAIN. If these words are written as in a dictionary, what
The total number of words is 60, and the 50th word is NAAIG.
The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is:
The probability of forming a 12-person committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor is (1)/(11)
From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and
95
The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must
1205
Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that
The eccentricity of the ellipse is (√2)/(2).
For n ≥ 2, let S_n denote the set of all subsets of 1, 2,, n with no two consecutive numbers. For example, 1, 3, 5 S_6, but 1, 2, 4 S_6. Then n(S_5) is equal to
13
There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that ca
210
What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these (i) four cards are of the same suit, (ii) four cards belong
The number of ways of choosing 4 cards from a pack of 52 playing cards is 270,725. (i) Four cards are of the same suit: 2,860 ways. (ii) Four cards belong to fo
Find the number of derangements of 1, 2,..., n using inclusion-exclusion principle
D_n = n! Σ_k=0^(n) ((-1)^k)/(k!) = n! ( 1 - (1)/(1!) + (1)/(2!) - (1)/(3!) + + (-1)^n (1)/(n!))
Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of
8925
The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is:
17280
The number of sequences of ten terms, whose terms are either 0, 1, or 2, that contain exactly five 1 ’s and exactly three 2 ’s is equal to:
2520
How many words, with or without meaning, each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?
2880
How many 2 digit even numbers can be formed from the digits 1, 2, 3, 4, 5 if the digits can be repeated?
10
how many word combinations can be made with the names in Vijayawada
151,200
The number of ways, in which the letters A, B, C, D, E can be placed in the 8 boxes of the figure below so that no row remains empty and at most one letter can
5760
Given 4 flags of different colours, how many different signals can be generated, if a signal requires the use of 2 flags one below the other?
12 different signals can be generated.
Let ^n C_r-1 = 28, ^n C_r = 56 and ^n C_r+1 = 70. Let A(4 cos t, 4 sin t), B(2 sin t, -2 cos t) and C(3r - n, r² - n - 1) be the vertices of a triangle ABC, whe
α = 20
In a group of 3 girls and 4 boys, there are two boys B_1 and B_2. The number of ways, in which these girls and boys can stand in a queue such that all the girls
144
From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is:
5148
If all the words with or without meaning made using all the letters of the word 'KANPUR' are arranged as in a dictionary, then the word at 440^(th) position in
PRKANU
Evaluate (n!)/(r!(n-r)!), when n = 5, r = 2.
10
How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
360
Find the number of arrangements of the letters of the word INDEPENDENCE. In how many of these arrangements, (i) do the words start with P (ii) do all the vowels
The number of arrangements of the letters of the word INDEPENDENCE is 1663200. (i) The number of arrangements that start with P is 138880. (ii) The number of ar
Three distinct numbers are selected randomly from the set 1,2,3,…,40. If the probability that the selected numbers are in an increasing G.P. is m/n, gcd(m,n)=1,
1981
Let S = p_1, p_2,, p_10 be the set of first ten prime numbers. Let A = S P, where P is the set of all possible products of distinct elements of S. Then the numb
5120
A committee of 3 persons is to be constituted from a group of 2 men and 3 women. In how many ways can this be done? How many of these committees would consist o
There are 10 ways to constitute a committee of 3 persons. Out of these, 6 committees would consist of 1 man and 2 women.
On her vacations Veena visits four cities (A, B, C and D) in a random order. What is the probability that she visits (i) A before B? (iii) A first and B last? (
(i) 1/2 (ii) 1/6 (iii) 1/12 (iv) 1/2 (v) 1/4

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