Advertisements
Advertisements
Question
Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
Advertisements
Solution
4-digit numbers are to be formed using the digits, 1, 2, 3, 4, and 5.
There will be as many 4-digit numbers as there are permutations of 5 different digits taken 4 at a time
Therefore, required number of 4 digit numbers =
5P4 = `(5!)/((5 - 4)!) = (5!)/(1!)`
= 1x 2 x 3 x 4 x 5 = 120
Among the 4-digit numbers formed by using the digits, 1, 2, 3, 4, 5, even numbers end with either 2 or 4.
The number of ways in which units are filled with digits is 2.
Since the digits are not repeated and the units place is already occupied with a digit (which is even), the remaining places are to be filled by the remaining 4 digits.
Therefore, the number of ways in which the remaining places can be filled is the permutation of 4 different digits taken 3 at a time.
Number of ways of filling the remaining places =
4P3 = `(4!)/((4 - 3)!) = (4!)/(1!)`
= 4 × 3 × 2 × 1 = 24
Thus, by multiplication principle, the required number of even numbers is
24 × 2 = 48
APPEARS IN
RELATED QUESTIONS
Find r if `""^5P_r = ""^6P_(r-1)`
In how many ways can the letters of the word PERMUTATIONS be arranged if the vowels are all together.
In how many ways can the letters of the word ASSASSINATION be arranged so that all the S’s are together?
Find x in each of the following:
In how many ways can three jobs I, II and III be assigned to three persons A, B and C if one person is assigned only one job and all are capable of doing each job?
If three six faced die each marked with numbers 1 to 6 on six faces, are thrown find the total number of possible outcomes ?
In how many ways can 7 letters be posted in 4 letter boxes?
In how many ways 4 women draw water from 4 taps, if no tap remains unused?
Write the number of ways in which 7 men and 7 women can sit on a round table such that no two women sit together ?
Write the number of words that can be formed out of the letters of the word 'COMMITTEE' ?
Write the number of all possible words that can be formed using the letters of the word 'MATHEMATICS'.
The number of words that can be formed out of the letters of the word "ARTICLE" so that vowels occupy even places is
The number of different signals which can be given from 6 flags of different colours taking one or more at a time, is
The number of ways in which the letters of the word 'CONSTANT' can be arranged without changing the relative positions of the vowels and consonants is
The number of arrangements of the letters of the word BHARAT taking 3 at a time is
The number of words that can be made by re-arranging the letters of the word APURBA so that vowels and consonants are alternate is
How many six-digit telephone numbers can be formed if the first two digits are 45 and no digit can appear more than once?
If (n+2)! = 60[(n–1)!], find n
In how many ways 5 boys and 3 girls can be seated in a row, so that no two girls are together?
How many 6-digit telephone numbers can be constructed with the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each numbers starts with 35 and no digit appear more than once?
- In how many ways can 8 identical beads be strung on a necklace?
- In how many ways can 8 boys form a ring?
Evaluate the following.
`(3! + 1!)/(2^2!)`
The greatest positive integer which divide n(n + 1) (n + 2) (n + 3) for all n ∈ N is:
If `""^(("n" – 1))"P"_3 : ""^"n""P"_4` = 1 : 10 find n
Determine the number of permutations of the letters of the word SIMPLE if all are taken at a time?
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?
Find the distinct permutations of the letters of the word MISSISSIPPI?
How many ways can the product a2 b3 c4 be expressed without exponents?
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 the vowels and consonants are alternative
Each of the digits 1, 1, 2, 3, 3 and 4 is written on a separate card. The six cards are then laid out in a row to form a 6-digit number. How many distinct 6-digit numbers are there?
The number of arrangements of the letters of the word BANANA in which two N's do not appear adjacently is ______.
If all permutations of the letters of the word AGAIN are arranged in the order as in a dictionary. What is the 49th word?
The number of signals that can be sent by 6 flags of different colours taking one or more at a time is ______.
The number of 5-digit telephone numbers having atleast one of their digits repeated 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 |
Let b1, b2, b3, b4 be a 4-element permutation with bi ∈ {1, 2, 3, .......,100} for 1 ≤ i ≤ 4 and bi ≠ bj for i ≠ j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1, b2, b3, b4 is equal to ______.
If 1P1 + 2. 2p2 + 3. 3p3 + ....... 15. 15P15 = qPr – s, 0 ≤ s ≤ 1, then q+sCr–s is equal to ______.
