Question

Seven digit numbers are formed using digits 1,2,3,4,5,6,7,8,9 without repetition. The probability of selecting a number such that product of any 5 consecutive digits is divisible by either 5 or 7 is p. Then 12p is equal to ?

Answer 1

Now, the sum of the digits from 1 through 9 is equal to 45, which happens to be a multiple of three. The only way to remove two digits while keeping the overall number divisible by three is to remove 2 digits that sum up to a multiple of 3. The possible multiples of 3 are: 3, 6, 9, 12, and 15. Now we must consider how many pairs of digits add up to one of these as follows.3: (1,2)6: (1,5), (2,4)9: (1,8), (2,7), (3,6), (4,5)12: (3,9), (4,8), (5,7)15: (6,9), (7,8)We can see that there are 12 pairs of digits that add up to a multiple of three. The 7-digit number can only be divisible by three if one of these pairs is removed. Thus out of the 36 pairs of digits that you could choose to remove, only 12 yield a number divisible by three.Thus, the probability of the 7-digit number being divisible by three is 12/36 or 1/3.

Answer 2

For a number to be divisible by three, the sum of its digits must be divisible by three. Thus, the order of the digits is irrelevant and we can just consider how many ways we can select any 7 digits. This can be found by

9C2=36 (this also equals 9C7 but 9C2 is more helpful as it represents choosing any two digits to omit)

Now, the sum of the digits from 1 through 9 is equal to 45, which happens to be a multiple of three. The only way to remove two digits while keeping the overall number divisible by three is to remove 2 digits that sum up to a multiple of 3. The possible multiples of 3 are: 3, 6, 9, 12, and 15. Now we must consider how many pairs of digits add up to one of these as follows.

3: (1,2)

6: (1,5), (2,4)

9: (1,8), (2,7), (3,6), (4,5)

12: (3,9), (4,8), (5,7)

15: (6,9), (7,8)

We can see that there are 12 pairs of digits that add up to a multiple of three. The 7-digit number can only be divisible by three if one of these pairs is removed. Thus out of the 36 pairs of digits that you could choose to remove, only 12 yield a number divisible by three.

Thus, the probability of the 7-digit number being divisible by five or seven
is 12/36 or 1/3.

hope it helps you