How many seven-digit positive integers satisfy the following conditions?

Question

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220

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The integer contains the digits 1,2,3,4,5,6 and 7 exactly once each.

The numbers 1,2 and3 appear in that order within the digits of the number. (They do not need to be consecutive digits.)

For example, two such integers are 7162534 and 1245673

Guest Aug 9, 2022

CPhill · Answer 1 · 2022-08-09T04:27:52

Not great at these counting problems but here's my best attempt

Choose any 3 of 7 positions for the digits 1,2,3 = C(7,3) = 35

For the other positions the other digits can be arranged in 4! = 24 ways

So....the total number of integers = 35 * 24 = 840