Answer:
72
Option c)
Explanation:
The total number of ways in which 5 digits can be arranged to form a number (without repeating digits) = 5! = 5 x 4 x 3 x 2 x 1 = 120
Number of ways in which 1 and 5 can appear together with 1 as the first digit and 5 as the second digit ie 15
15 can be in the first two places, second and third place, third and fourth place and fourth and fifth place
Using x as the other digits in the number the combinations possible with 1 and 5 adjacent to each other for each combination of the other digits
There are a total of 4 ways in which 1 and 5 can appear together for each combination of the other digits
1 5 x x x
x 1 5 x x x
x x 1 5 x x
x x x 1 5
For just the first two places we have the following 6 possibilities
1 5 2 3 4
1 5 2 4 3
1 5 3 2 4
1 5 3 4 2
1 5 4 2 3
1 5 4 3 2
Since there are a total of 4 possible combinations each resulting in 6 ways a number with 1 in the first place and 5 in the second place can occur the total such combinations is 4 x 6 = 24
Now 1 and 5 can appear together as 5 1
This would result in another 24 combinations
So total number of combinations where 1 and 5 can occur either as 1 5 or 5 1 = 2 x 24 = 48
Since total number of possible 5 digit numbers is 120, the numbers that do not have 1 and 5 adjacent = 120 - 48 = 72