Answer:
a)286 ways
b)1,037,836,800 ways
Explanation:
a. How many ways are there to choose 10 players to play the game?
We have to take note of a key word here which is CHOOSE. For question a, order does not matter.
Hence, we use the combination formula. This is given as:
C(n, r) = nCr = n!/r! (n - r)!
n = 13, r = 10
13C10 = 13!/10! (13 - 10)!
= 13!/ 10! × (3!)
= 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 / (10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) × (3 × 2 × 1)
= 1716/6
= 286 ways.
b. How many ways are there to assign the 10 different positions by selecting players from the 13 who show up?
For question b as well, we take note of a key word which is ASSIGN. For question b, order is very important.
Therefore, the formula we use is the permutation formula.
P(n, r) = nPr = n!/(n - r)!
n = 13, r = 10
13P10 = 13!/ (13 - 10)!
= 13!/ 3!
= 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 / (3 × 2 × 1)
= 1,037,836,800 ways