Lesson 6 of 11 · 6 min
Permutations with repeated objects
NCERT §6.3.4
The house banner spells BALLOON in cut-out letters, and the art club wants to know how many different words the 7 letters can make. Two L's and two O's look exactly alike, so some arrangements are the same word.
The lesson in notes
In short
If some objects are alike, swapping them does not give a new arrangement, so dividing by the number of such swaps removes the repeats.
The number of arrangements of n objects of which p₁ are alike of one kind, p₂ alike of another kind, and so on, is n!/(p₁! p₂! … pₖ!).
BALLOON has 7 letters with L twice and O twice, so it has 7!/(2! × 2!) = 5040/4 = 1260 arrangements.
Pennants: 3 red, 2 blue and 1 green pennant in a row, same colours alike, give 6!/(3! × 2! × 1!) = 720/12 = 60 patterns.
SPORTS has S twice: 6!/2! = 360 arrangements. Arrangements of BALLOON that start with B: fix B and arrange A, L, L, O, O, N, giving 6!/(2! × 2!) = 180.
Check with labels: tag the two Ls as L₁ and L₂. Each arrangement with plain Ls matches 2! labelled ones, which is why we divide by 2!.