The question, as stated in the title, is:
Find the total number of 4 digit numbers in which the digits are in descending order.
Answer given: $\displaystyle {10 \choose 4} \text{ or } 210.$
I've tried listing doing the possibilities of these numbers:
- Numbers starting with $9$:
$9876,9875,…,9871,9870; 9765,9764,…9761,9760; 9654,9653,…,9651,9650;9543,9542,9541;9432,9431,9430,9420;9321,9320,… \text{and so on.}$
I could go on and count all the numbers in this way, but it's quite tedious and not very optimal. How do I use combinatorics here?
I would prefer a complete explanation.
EDIT: All the digits have to be different, as pointed out by @TonyK. I appologize for not clarifying earlier.
Best Answer
Hint: Such a number is completely determined once you choose which four digits appear in it.