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
Evaluate 4! – 3!
Compute `(8!)/(6! xx 2!)`
if `1/(6!) + 1/(7!) = x/(8!)`, find x
Find r if `""^5P_r = 2^6 P_(r-1)`
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 in a group of n distinct objects, the number of arrangements of 4 objects is 12 times the number of arrangements of 2 objects, then the number of objects is
English alphabet has 11 symmetric letters that appear same when looked at in a mirror. These letters are A, H, I, M, O, T, U, V, W, X and Y. How many symmetric three letters passwords can be formed using these letters?
How many five digits telephone numbers can be constructed using the digits 0 to 9 If each number starts with 67 with no digit appears more than once?
In how many ways 5 boys and 3 girls can be seated in a row, so that no two girls are together?
- In how many ways can 8 identical beads be strung on a necklace?
- In how many ways can 8 boys form a ring?
Suppose 8 people enter an event in a swimming meet. In how many ways could the gold, silver and bronze prizes be awarded?
A student appears in an objective test which contain 5 multiple choice questions. Each question has four choices out of which one correct answer.
What is the maximum number of different answers can the students give?
In how many ways can the letters of the word SUCCESS be arranged so that all Ss are together?
Choose the correct alternative:
If `""^(("n" + 5))"P"_(("n" + 1)) = ((11("n" - 1))/2)^(("n" + 3))"P"_"n"`, then the value of n are
In how many ways can 5 children be arranged in a line such that two particular children of them are always together
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 ______.
Five boys and five girls form a line. Find the number of ways of making the seating arrangement under the following condition:
| C1 | C2 |
| (a) Boys and girls alternate: | (i) 5! × 6! |
| (b) No two girls sit together : | (ii) 10! – 5! 6! |
| (c) All the girls sit together | (iii) (5!)2 + (5!)2 |
| (d) All the girls are never together : | (iv) 2! 5! 5! |
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 ______.
Number of words from the letters of the words BHARAT in which B and H will never come together is ______.
