If a set A has 12 elements. Find the number of subsets of A having
(i) 4 elements     (ii) atleast 3 elements       (ii) atmost 3 elements

Number of elements of set A=12

Number of subsets of A having

(i) 4 elements is ${ }^{12}{C}_4=\frac{12\times 11\times 10\times 9}{4\times 3\times 2\times 1}=495$

(ii) at least 3 elements= $$\begin{gathered} { }^{12}{C}_3+{ }^{12}{C}_4+{ }^{12}{C}_5+.....+{ }^{12}{C}_{12} \\ =2^{12}-{(}^{12}{C}_0{+}^{12}{C}_1{+}^{12}C2) \\ =4096-(1+12+1) \\ =4017 \end{gathered}$$

(iii)Atmost 3 elements=$$\begin{gathered} { }^{12}{C}_0{+}^{12}{C}_1{+}^{12}{C}_2+{ }^{12}{C}_3 \\ =1+12+66+220 \\ =299 \end{gathered}$$