# Problem of the Day: 12/24/12

Santa’s elves are busy making Christmas gifts at the North Pole! They have to make 3 billion gifts in time for midnight on Christmas Eve – but luckily, they have 200,000 elves working at any given time. If each elf can make a gift in one minute (they’re highly trained), then on what day in December would the elves have to start making gifts?

Solution to yesterday’s problem:
To find the probability that five specific students are chosen out of 20, but the order of the students doesn’t matter, we can use combinations.

The number of possible ways that a group of five can be picked out of 20 is
$_{20}C_5=\frac{20!}{5!(20-5)!}$
To simplify this easily, we can rewrite the factorials as multiplications, then cancel:
$\frac{20\times19\times18\times17\times16\times15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1}{5\times4\times3\times2\times1(15\times14\times13\times12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1)}$
Canceling,
$\frac{20\times19\times18\times17\times16}{5\times4\times3\times2\times1}$
Now we can do another round of dividing by common factors, like 20 and 5, 16 and 4 and 2, etc.
$4\times19\times6\times17\times2=15504$
So the chance that these 5 particular students are chosen is incredibly small: $\frac{1}{15504}$.