Up-and-down Number

In an up-and-down counting number, the digits increase to a maximum digit and then decrease. This maximum is not the first or last digit. (A few examples: 1247321 is an up-and-down counting number; 12477321 is not an up-and-down counting number; 13557321 is not an up-and-down counting number.) How many different 4-digit up-and-down numbers are there in which the maximum digit is 6 and at least one of the digits is a 3?

Multiples of 33

What's the total number of different ways that the blanks of "__ 3 __ 9" can be filled in so that the resulting four-digit number is a multiple of 33?

Pick a Number

An integer is chosen at random from the set {21, 22, 23, ..., 81}. What's the probability that the chosen number is one more than a multiple or 4 or one more than a multiple of 5?

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A similar problem was given in Assignment 2 of Charlotte Math Meetup:

An integer is chosen at random from the set {41, 42, 43, ..., 67}. What's the probability that the chosen number is one more than a multiple or 4 or one more than a multiple of 5? (Please express the answer as a decimal number rounded to hundredth.)