Advertisements
Advertisements
Question
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 of these 6-digit numbers are divisible by 4?
Advertisements
Solution
In order to get the 6-digit number divisible by 4
The last two digits must be divisible by 4
∴ The last two digits should be 12 or 24 or 32
| 24 | ||||
| 1 | 2 | 3 | 4 | 5 |
Let the last box be filled with 24.
The remaining 4 boxes can be filled with the remaining digits
1, 1, 3, 3 in `(4!)/(2! xx 2!)` ways.
| 12 | ||||
| 1 | 2 | 3 | 4 | 5 |
Let the last box be filled with 12.
The remaining 4 boxes can be filled with the remaining digits.
1, 3, 3, 4 in `(4!)/(2!)` ways
| 32 | ||||
| 1 | 2 | 3 | 4 | 5 |
Let the last box be filled with 32.
The remaining 4 boxes can be filled with the remaining digits
The total number of 6 digit numbers which are divisible by 4 is
= `(4!)/(2! xx 2!) + (4!)/(2!) + (4!)/(2!)`
= `(1 xx 2 xx 3 xx 4)/(1 xx 2 xx 1 xx 2) + (1 xx 2 xx 3 xx 4)/(1 xx 2) + (1 xx 2 xx 3 xx 4)/(1 xx 2)`
= 6 + 12 + 12
= 30
∴ Required number of 6-digit numbers = 30
APPEARS IN
RELATED QUESTIONS
How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
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!
Evaluate each of the following:
P(6, 4)
The number of five-digit telephone numbers having at least one of their digits repeated is
The number of words from the letters of the word 'BHARAT' in which B and H will never come together, is
The number of arrangements of the word "DELHI" in which E precedes I 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 ways to arrange the letters of the word CHEESE are
A 5-digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways in which this can be done is
In a room there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amounts of illumination is
In how many ways 5 boys and 3 girls can be seated in a row, so that no two girls are together?
Evaluate the following.
`((3!)! xx 2!)/(5!)`
For all n > 0, nC1 + nC2 + nC3 + …… + nCn is equal to:
Determine the number of permutations of the letters of the word SIMPLE if all are taken at a time?
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 of these 6-digit numbers are even?
If the letters of the word GARDEN are permuted in all possible ways and the strings thus formed are arranged in the dictionary order, then find the ranks of the words
GARDEN
Ten different letters of alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated is ______.
There are 10 persons named P1, P2, P3, ... P10. Out of 10 persons, 5 persons are to be arranged in a line such that in each arrangement P1 must occur whereas P4 and P5 do not occur. Find the number of such possible arrangements.
The number of 5-digit telephone numbers having atleast one of their digits repeated is ______.
