Lesson 4 of 11 · 11 min
Permutations of distinct objects
NCERT §6.3, §6.3.1, §6.3.3
Twelve students want the posts of captain and vice-captain of the sports team. Later, 4 of the 7 fastest sprinters will run the 4 legs of the relay. In both jobs the order of the chosen people matters.
The lesson in notes
In short
A permutation is an arrangement of objects in a definite order, using some or all of them. Changing the order gives a different permutation.
nPr counts ordered arrangements of r things drawn from n different things with no repeats: nPr = n(n − 1)(n − 2) … (n − r + 1) = n!/(n − r)!, for 0 ≤ r ≤ n.
Special cases: nPn = n!/0! = n!, and nP0 = 1, since there is exactly one way to arrange nothing.
If repetition is allowed, each of the r places has all n choices, so the number of arrangements is nʳ.
Sports day: a captain and a vice-captain are chosen from 12 students, one post each. The order matters (the two posts differ), so the count is 12P2 = 12 × 11 = 132.
Relay: 4 of 7 sprinters are placed in running order as first, second, third and fourth legs: 7P4 = 7 × 6 × 5 × 4 = 840 line-ups.
Words from the letters of MEDAL (5 different letters): 3-letter words 5P3 = 60, 5-letter words 5! = 120, and 3-letter words with repetition allowed 5³ = 125.
Solving for n: if nP2 = 56, then n(n − 1) = 56 = 8 × 7, so n = 8.