Desperate to know....

lechefrene wrote:...how many 5 digit numbers can be formed from a set of seven digits, without repeating digits, and not transposing numbers? For example, if I have the digits 1234567, how many five digit numbers can be formed? 12345, 23456, 34567, etc. are good numbers, but 54321 would not be valid as it is the same as 12345...also, 11234 is invalid as it repeats the digit 1. Please answer ASAP, and thanks so much in advance! ~lechefrene~

I'll explain in three different ways:

The digits in the number must be in ascending order- that is, they must be in one particular order. If we could put the five digits we select in any order, the answer would be 7*6*5*4*3. However, for any set of five digits we choose, only one of the 5! orders we can put five digits in is acceptable. So we need to divide this by 5!, which gives 21.

Equivalently- once you choose a set of five digits, you must put them in ascending order to make an acceptable number, so the answer will be 7C5 = 7*6/2 = 21.

Or, you can just think of how many ways you could remove two digits from 1234567. For any pair you remove, you will get a different 5-digit number with digits in ascending order, and we can choose a pair of digits in 7C2 ways, or 21 ways.