Delivery Routes
A distribution warehouse has 5 drones to deliver packages between 12 different routes. How many drone/route combinations are possible?
Expand Hint
Unlike a permutation where a particular sequence order is considered, a combination is needed to find the number of routes because order is not important.
Hint 2
$$$C_{(n,r)}=\frac{P_{(n,r)}}{r!}=\frac{n!}{[r!(n-r)!]}$$$
where
$$C_{n,r}$$
is the number of different combinations of
$$n$$
distinct objects taken
$$r$$
at a time, and
$$P$$
is the number of different permutations.
Unlike a permutation where a particular sequence order is considered, a combination is needed to find the number of groups because order does not impact how pairings are formed:
$$$C_{(n,r)}=\frac{P_{(n,r)}}{r!}=\frac{n!}{[r!(n-r)!]}$$$
where
$$C_{n,r}$$
is the number of different combinations of
$$n$$
distinct objects taken
$$r$$
at a time, and
$$P$$
is the number of different permutations.
Thus, the number of possible drone/route formations based on the problem statement is:
$$$C_{(12,5)}=\frac{12!}{5![(12-5)!]}=\frac{12!}{5!(7!)}=\frac{12 \cdot 11\cdot 10\cdot 9\cdot 8\cdot 7!}{5 \cdot 4\cdot 3\cdot 2\cdot 1\cdot (7!)}=\frac{95,040}{120}=792$$$
792
Time Analysis
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