# Combination – Properties of ⁿCr

## Combination – Properties of ⁿCr

Each of the different groups of selections, which can be made by taking some or all of a number of given things or objects at a time is called a combination. In combination, order of appearance of things is not taken into account

Example: Three groups can be made with three different objects a, b, c taking two at a time, i.e., ab, bc, ac.

Here, ab and ba are the same group. It is also clear that for each combination (selection or group) of two things, number of permutations (arrangements) is 2!.  For example, for combination ab there are two permutations, i.e., ab and ba.

Number of combinations of n different things taking r at a time (n ≥ r):

$$^{n}{{C}_{r}}=\frac{n!}{r!(n-r)!}$$.

Proof: Let the number of combinations of n different things taken r at time be ⁿCr.

Now, each combination consists of r different things and these r things can be arranged among themselves in r! ways.

Thus, for one combination, number of arrangements is r! ⁿCr … (1)

But number of permutations of n different things taken r at a times is ⁿPr … (2)

From (1) and (2) we get

$$r!{{\ }^{n}}{{C}_{r}}{{=}^{n}}{{P}_{r}}=\frac{n!}{(n-r)!}$$.

$$^{n}{{C}_{r}}=\frac{n!}{r!(n-r)!}$$.

Properties of ⁿCr:

1. ⁿCr = ⁿCn-r.

Proof: $$^{n}{{C}_{r}}=\frac{n!}{r!(n-r)!}$$ … (1)

$$^{n}{{C}_{n-r}}=\frac{n!}{(n-r)!(n-(n-r)!}$$.

$$^{n}{{C}_{n-r}}=\frac{n!}{(n-r)!(r)!}$$ … (2)

From equation (1) and (2)

$$^{n}{{C}_{n-r}}=\frac{n!}{(n-r)!(r)!}{{=}^{n}}{{C}_{r}}$$.

Hence proved ⁿCr = ⁿCn-r.

2. ⁿCx = ⁿCy, then either x = y or x + y = n.

Proof: ⁿCx = ⁿCy = ⁿCn-y … (1)

(∵ ⁿCr = ⁿCn-r)

From (1)

x = y

x = n – y (or) x + y = n.