Combination Formula

Combination Formula

A selection that can be formed by taking some or all finite set of things (or objects) is called a combination. Formation of a combination by taking ‘f’ elements from a finite set A containing ‘n’ elements means picking up f-elements subset of A ($n \geqslant f$).
The number of combinations of n dissimilar things taken ‘k’ at a time or choosing k objects or things from n objects is denoted by
$^n{C_k}\,\,{\text{or }}C\left( {n,k} \right)\,\,{\text{or }}C\left( {\begin{array}{*{20}{c}}
  n \\
  k
\end{array}} \right)\,\,\,{\text{or }}\left( {\begin{array}{*{20}{c}}
  n \\
  k
\end{array}} \right).$
Also $^n{C_k} = \frac{{n!}}{{k!\left( {n - k} \right)!}}$

Example:
If there are 12 persons in a party, and if each two of them shake hands with each other, the number of handshakes in the part is _____
Solution: It is to note that, when two persons shake hands, it is counted as one handshake. The total numbers of handshakes is some as the number of ways of selecting 2 persons among 12 persons.
Thus,
$^{12}{C_2} = \frac{{12!}}{{10!2!}} = 66$.

Question:
No. of ways of selecting 2 girls and 3 boys from 3 girls and 5 boys is
Options:
(a) 20
(b) 24
(c) 30
(d) 48