Question

There are 5 mangoes, 4 apples and 3 bananas in a bag.in how many ways can atleast 2 fruits be selected if:

i)all fruits are different.

ii)all mangoes are similar, all apples are similar and all bananas are similar.

Answer 1

$$\begin{gathered} \mathrm{There}\mathrm{are}5\mathrm{mangoes},4\mathrm{apples}\mathrm{and}3\mathrm{bananas} \\ \mathrm{So}\mathrm{if}\mathrm{all}\mathrm{are}\mathrm{different}\mathrm{then}\mathrm{number}\mathrm{of}\mathrm{ways}\mathrm{we}\mathrm{can}\mathrm{choose}2\mathrm{and}\mathrm{more}\mathrm{fruit}\mathrm{is} \\ \mathrm{As}\mathrm{here}\mathrm{total}\mathrm{number}\mathrm{of}\mathrm{fruits}=12 \\ {=}^{12}{\mathrm{C}}_2{+}^{12}{\mathrm{C}}_3{+}^{12}{\mathrm{C}}_4{+}^{12}{\mathrm{C}}_5{+}^{12}{\mathrm{C}}_6{+}^{12}{\mathrm{C}}_7{+}^{12}{\mathrm{C}}_8{+}^{12}{\mathrm{C}}_9{+}^{12}{\mathrm{C}}_{10}{+}^{12}{\mathrm{C}}_{11}{+}^{12}{\mathrm{C}}_{12} \\ \mathrm{Simplify}\mathrm{this}\mathrm{for}\mathrm{your}\mathrm{own}\mathrm{practice}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{answer}. \\ \mathrm{b}) \\ \mathrm{when}\mathrm{all}\mathrm{the}\mathrm{mangoes}\mathrm{are}\mathrm{similar},\mathrm{all}\mathrm{the}\mathrm{apples}\mathrm{are}\mathrm{similar}\mathrm{and}\mathrm{all}\mathrm{the}\mathrm{bananas}\mathrm{are}\mathrm{similar}, \\ =2\mathrm{apples}\mathrm{from}\mathrm{similar}+2\mathrm{mangoes}\mathrm{from}\mathrm{similar}+2\mathrm{banana}\mathrm{from}\mathrm{similar}+1\mathrm{apple}1\mathrm{mango}+1\mathrm{apple},\mathrm{a}\mathrm{banana} \\ +1\mathrm{banan},1\mathrm{mango} \\ {=}^4{\mathrm{C}}_2{+}^5{\mathrm{C}}_2{+}^3{\mathrm{C}}_2{+}^5{\mathrm{C}}_1{\times }^4{\mathrm{C}}_1{+}^3{\mathrm{C}}_1{\times }^4{\mathrm{C}}_1{+}^5{\mathrm{C}}_1{\times }^3{\mathrm{C}}_1 \\ \mathrm{Simplify}\mathrm{for}\mathrm{your}\mathrm{own}\mathrm{practice}\mathrm{to}\mathrm{get}\mathrm{the}\mathrm{answer}. \end{gathered}$$