Exercise 14.3
Page no 14.36
Type 1
Question 1
If $\frac{{ }^{n} P_{4}}{{ }^{n-1} P_{4}}=\frac{5}{3}, n>4$, find $n$
Question 2
Find $r$ if
(i) ${ }^{10} P_{r}=2,{ }^{9} P_{r}$
(ii) ${ }^{5} P_{r}={ }^{6} P_{r-1}$
(iii) ${ }^{5} P_{r}=2 .{ }^{6} P_{r-1}$
Question 3
If ${ }^{\prime \prime} P_{4}=12 \times{ }^{n} P_{2}$, find $n$.
Question 4
If ${ }^{n} P_{5}=20 \times{ }^{n} P_{3}$, find the value of $n$.
Question 5
If ${ }^{n} P_{4}:{ }^{n+1} P_{4}=3: 4$, find $n$.
Question 6
(i) If ${ }^{20} P_{r}=6840$, find $r$.
(ii) If ${ }^{12} P_{r}=11880$, find $r$.
Question 7
Prove that ${ }^{10} P_{3}={ }^{9} P_{3}+3 \cdot{ }^{9} P_{2}$
Question 8
If ${ }^{k+5} P_{k+1}=\frac{11(k-1)}{2}, k+3 P_{k}$, find $k$
Question 9
If ${ }^{22} P_{r+1}:{ }^{22} P_{r+2}=11: 52$, find $k$.
Question 10
If $m+n P_{2}=90$ and $m-n P_{2}=30$, find $m$ and $n$.
Question 11
If ${ }^{56} P_{r+6}:{ }^{54} P_{r+3}=30800: 1$, find $r$.
Type 2
Question 12
How many four digit numbers are there, with no digit repeated ?
Question 13
How many even number of three digits each can be made with the digits 1,2,3,4, 6,7 if no digit is repeated?
Question 14
How many numbers of four digits can be formed with the digits 1,2,4,5,7 no digit being repeated?
Page no - 14.37
Question 15
How many numbers of 5 digits can be formed with the digits $0,1,2,3,4 ?$
Question 16
Find the number of 4-digit numbers that can be formed using the digits $1,2,3$. 4. 5 if no digit is used more than once in a number. How many of these numbers will be even ?
Question 17
(i) How many numbers between 100 and 1000 can be formed with the digits, $1,2,3,4,5,6,7$; no digit being repeated ?
(ii) How many numbers lying between 100 and 1000 can be formed with the digits, $0,1,2,3,4,5$ if the repetitions of the digits is not allowed?
Question 18
How many numbers each lying between 100 and 1000 can be formed with the digits $2,3,4,0,8,9$; no digit being repeated ?
Question 19
Find the total numbers of nine digit numbers which have all different digits.
Question 20
(i) How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed ?
(ii) How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated ?
Question 21
How many of the natural numbers from 1 to 1000 have none of their digits repeated?
Question 22
How many numbers each lying between 1000 and 10000 can be formed with the digits $0,1,2,3,4,5$; no digit being repeated ?
Question 23
How many different numbers greater than 5000 can be formed with the digits $0,1,5,9$, no digit being repeated ?
Question 24
Find the number of numbers lying between 300 and 4000 that can be formed with the digits $0,1,2,3,4,5$; no digit being repeated.
Question 25
If repetition of digits is not allowed how many numbers of four digits divisible by 5 can be formed with the digits $0,4,5,6,7$ ?
Question 26
If repetition of digits is not allowed how many different numbers of 6 digits each can be formed with the digits $4,5,6,7,8,9$ ? How many of them are not divisible by 5 .
Question 27
How many even numbers of 5 digits without repetition can be formed with the digits $1,2,3,4$ and 5 .
Question 28
How many numbers less than 1000 and divisible by 5 can be formed in which no digit occurs more than once in the same number?
Question 29
Find how many numbers between 100 and 999 can be formed with the digits $0,4,5,6,7,8$; no digit being used more than once. How many of them are odd ?
Question 30
(i) Find the number of numbers of six digits without repetition formed with the digits $1,2,3,4,5,6$ in which 5 always occurs in the ten's place.
(ii) How many 6-digit numbers can be formed from the digits $0,1,3,5,7$ and 9 which are divisible by 10 and no digit is repeated.
Question 31
A number of four different digits without repetition is formed by using the digits $1,2,3,4,5,6,7$. Find :
(i) How many such numbers can be formed?
(ii) How many of them are greater than $3400 ?$
Question 32
Find the number of numbers of 4 digits without repetition formed with the digits $1,2,3,4,5$ in which 4 occurs in the thousand's place and 5 occurs in the unit's place.
Page no 14.38
Question 33
Find the number of positive integers which can be formed by using any number of digits from. 0,1,2,3,4,5 But using each digit not more than once in each number . How many of these integers are greater than 3000 ?
Question 34
How many different numbers can be formed with the digits $1,3,5,7$ and 9 . when taken all at a time and what is their sum ?
Question 35
Find the sum of all the 4 digit numbers that can be formed with the digits 0 . $2,3,5$.
Type 3
Question 36
A servant has to post 5 letters and there are 4 letters boxes. In how many ways can he post the letters ?
Question 37
In how many ways can three prizes be given away to 5 students when each student is eligible for any of the prizes?
Question 38
In how many ways can $n$ things be given to $p$ persons, when each person can get any number of things $(n>p)$.
Question 39
Find the number of functions that can be defined from $A$ to $B$ if number of distinct elements in $A$ and $B$ are $m$ and $n$ respectively.
Question 40
In how many different ways the following 5 prizes be distributed among 10 students ? First and second in Mathematics; first and second in Physics and first in Hindi.
Question 41
There are stalls for 12 animals in a ship. In how many ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 12 ?
Question 42
In how many ways 5 delegates can be put in 6 hotels of a city if there is no restriction ?
Question 43
Find the number of numbers of 5 digits that can be formed with the digits $0,1,2,3$ and 4 if repetition of digits is allowed.
Question 44
In how many ways 6 rings of different types can be had in 4 given fingers of a hand ?
Question 45
Find the number of numbers of 4 digits greater than 3000 that can be formed with the digits $0,1,2,3,4$ and 5 if repetition of digits is allowed.
Question 46
In a town the car plate numbers contain only three or four digits, not containing the digit 0 . What is the maximum number of cars that can be numbered?
Question 47
In how many ways can a ten question multiple choice examination be answered if there are four choices $a, b, c$ and $d$ to each question ? If no two consecutive questions are answered the same way, how many ways are there?
[HOTS]
Question 48
Find the number of numbers of four digits that can be made from the digits $0,1,2,3,4,5$ if digits can be repeated in the same number. How many of these numbers have at least one digit repeated?
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