Question

How many integers larger than 3, 000, 000 can be formed by arranging the digits
1, 1, 2, 5, 5, 6, 9? How many can we form so that the digit is immediately followed
by a larger digit?

Answer 1

We have 2 parts in the above question.

a) Note that 3,000,000 has got 6 digits after 3.

We have a total of 7 digits given here.

Therefore for integers greater than 3,000,000, we need the first digit to be one of the digit from 5, 6 or 9

Therefore the number of integers in all combinations here is computed as:

• First digit is 5: Number of digits left are: 5, 1, 1, 2, 6, 9. Therefore number of total arrangements possible here is computed as: 6! / 2 = 360. Note that we divided by 2 because there are two 1s here.
• First digit is 6: Number of digits left are: 1, 1, 2, 5, 5, 9. Therefore number of total arrangements possible here is computed as: 6! / 2*2 = 180
• First digit is 9: Number of digits left are: 1, 1, 2, 5, 5, 6. Therefore number of total arrangements possible here is computed as: 6! / 2*2 = 180

Therefore total numbers possible here: 360 + 180*2 = 720

Therefore 720 numbers are possible here.

b) Given that the digit should be immediately followed by a larger digit, no such number is possible here, this is because we have pair of same numbers which we would never be able to fit 2 equal numbers when the digit has to be followed by a larger digit.