Number of Combinations of a 4 Digit Lock
One day, no one in the office remembers the password for a 4-digit lock on a utility room.
Initially, we thought the number of combinations to figure it out brutally is \(P(10, 4) = \frac{10!}{(10 - 4)!} = 10 \times 9 \times 8 \times 7 = 5040\) by using the formula of permutations.
Then someone remembers that the lock allows duplicates. So the number of combinations increases to \(10 \times 10 \times 10 \times 10 = 10000\) since there are \(10\) options for each digit.
Later someone remembers that the order of numbers doesn't matter. Then we are dealing with combinations. The number of combinations is reduced to $$ C(10, 4) = \frac{10!}{4! (10 - 4)!} = 210 $$ The number is much lower to deal with.
Further reading: Permutations and combinations