Simulation · Maths · Class 11

Fill the places, then count

From the lesson Permutations of distinct objects in Permutations and Combinations. Change the values and watch what happens.

The idea behind it

NCERT §6.3, §6.3.1, §6.3.3

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.

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