Advertisements
Advertisements
Question
How many natural numbers not exceeding 4321 can be formed with the digits 1, 2, 3 and 4, if the digits can repeat?
Solution
Case I: Four-digit number
Total number of ways in which the 4 digit number can be formed =`4xx4xx4xx4=256`
Now, the number of ways in which the 4-digit numbers greater than 4321 can be formed is as follows:
Suppose, the thousand's digit is 4 and hundred's digit is either 3 or 4.
∴ Number of ways =`2xx4xx4=32`
But 4311, 4312, 4313, 4314, 4321 (i.e. 5 numbers) are less than or equal to 4321.
∴ Remaining number of ways =`256-(32-5)=229`
Case II: Three-digit number
The hundred's digit can be filled in 4 ways.
Similarly, the ten's digit and the unit's digit can also be filled in 4 ways each. This is because the repetition of digits is allowed.
∴ Total number of three-digit number =`4xx4xx4=64`
Case III: Two-digit number
The ten's digit and the unit's digit can be filled in 4 ways each. This is because the repetition of digits is allowed.
∴ Total number of two digit numbers `4xx4=16`
Case IV: One-digit number
Single digit number can only be four.
∴ Required numbers = 229 + 64 + 16 +4 = 313
APPEARS IN
RELATED QUESTIONS
Evaluate 8!
Is 3! + 4! = 7!?
Compute `(8!)/(6! xx 2!)`
How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
From a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position?
Find r if `""^5P_r = 2^6 P_(r-1)`
How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if
(i) 4 letters are used at a time,
(ii) all letters are used at a time,
(iii) all letters are used but first letter is a vowel?
In how many ways can the letters of the word PERMUTATIONS be arranged if the vowels are all together.
Which of the following are true:
(2 +3)! = 2! + 3!
How many 5-digit telephone numbers can be constructed using the digits 0 to 9. If each number starts with 67 and no digit appears more than once?
Find the number of ways in which one can post 5 letters in 7 letter boxes ?
In how many ways can 4 prizes be distributed among 5 students, when
(i) no student gets more than one prize?
(ii) a student may get any number of prizes?
(iii) no student gets all the prizes?
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which the hall can be illuminated ?
Evaluate each of the following:
Evaluate each of the following:
6P6
Evaluate each of the following:
P(6, 4)
Write the number of numbers that can be formed using all for digits 1, 2, 3, 4 ?
If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is
The number of ways in which 6 men can be arranged in a row so that three particular men are consecutive, is
The number of ways in which the letters of the word ARTICLE can be arranged so that even places are always occupied by consonants is
Find the number of arrangements that can be made out of the letters of the word “ASSASSINATION”.
The possible outcomes when a coin is tossed five times:
The number of words with or without meaning that can be formed using letters of the word “EQUATION”, with no repetition of letters is:
A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.
How will the answer change if each question may have more than one correct answers?
8 women and 6 men are standing in a line. In how many arrangements will all 6 men be standing next to one another?
A coin is tossed 8 times, how many different sequences containing six heads and two tails are possible?
How many strings are there using the letters of the word INTERMEDIATE, if vowels are never together
Choose the correct alternative:
The product of r consecutive positive integers is divisible b
How many words can be formed with the letters of the word MANAGEMENT by rearranging them?
Find the number of permutations of n different things taken r at a time such that two specific things occur together.
Find the number of different words that can be formed from the letters of the word ‘TRIANGLE’ so that no vowels are together
In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. How many telephone numbers have all six digits distinct?
A five-digit number divisible by 3 is to be formed using the numbers 0, 1, 2, 3, 4 and 5 without repetitions. The total number of ways this can be done is ______.
The number of words which can be formed out of the letters of the word ARTICLE, so that vowels occupy the even place is ______.
How many words (with or without dictionary meaning) can be made from the letters of the word MONDAY, assuming that no letter is repeated, if
| C1 | C2 |
| (a) 4 letters are used at a time | (i) 720 |
| (b) All letters are used at a time | (ii) 240 |
| (c) All letters are used but the first is a vowel | (iii) 360 |
The number of three-digit even numbers, formed by the digits 0, 1, 3, 4, 6, 7 if the repetition of digits is not allowed, is ______.
