**Combination – Properties of ⁿC**_{r}

_{r}

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 ⁿC_{r}.

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! ⁿC_{r} … (1)

But number
of permutations of n different things taken r at a times is ⁿP_{r} …
(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 ⁿC _{r}:**

**1. ⁿC _{r} = ⁿC_{n-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
ⁿC_{r} = ⁿC_{n-r}.

**2. ⁿC _{x} = ⁿC_{y},
then either x = y or x + y = n.**

**Proof: **ⁿC_{x} = ⁿC_{y} = ⁿC_{n-y}
… (1)

(∵ ⁿC_{r} = ⁿC_{n-r})

From (1)

x = y

x = n – y (or) x + y = n.