Lesson 7 of 11 · 6 min
Combinations
NCERT §6.4
The relay needs a squad of 4 from the 7 fastest sprinters, but the running order will be decided later. Now only the group matters, not who runs first.
The lesson in notes
In short
A combination is a selection in which order does not matter: choosing A and B is the same as choosing B and A.
nCr = n!/(r!(n − r)!), for 0 ≤ r ≤ n. It links to permutations by nPr = nCr × r!, because each selection of r objects can be put in order in r! ways.
Relay: the team of 4 from 7 sprinters can be picked in 7C4 = 35 ways. Each team has 4! = 24 running orders, and 35 × 24 = 840 = 7P4.
Handshakes: 10 house captains each shake hands once with every other: 10C2 = 45 handshakes.
Points on a circle: 8 points give 8C2 = 28 chords and 8C3 = 56 triangles, since no three of them lie on a line.
nC0 = 1 and nCn = 1: there is one way to choose nothing and one way to choose everything. A committee of 3 from 6 volunteers can be picked in 6C3 = 20 ways.