4 players a,b,c,d are playinf with a ball giving catches to each other .if a start giving catches to any of the players b,c,d then find the no of ways in which the ball can return to a when it is known that player cant catch the ball more than 2 times???

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{a} \mathrm{has} 3 \mathrm{options}, \mathrm{d} \mathrm{or} \mathrm{b} \mathrm{or} \mathrm{c} \\ \mathrm{Now} \mathrm{they} \mathrm{return} \mathrm{it} \mathrm{to} \mathrm{a} \mathrm{which} \mathrm{is} 1 \mathrm{options} \\ \mathrm{They} \mathrm{give} \mathrm{it} \mathrm{to} \mathrm{any} \mathrm{other} \mathrm{except} \mathrm{a} \mathrm{with} 2 \mathrm{options} \mathrm{and} \mathrm{that} \mathrm{person} \mathrm{gives} \\ \mathrm{it} \mathrm{to} \mathrm{third} \mathrm{person} \mathrm{who} \mathrm{returns} \mathrm{it} \mathrm{to} \mathrm{a} \mathrm{which} \mathrm{is} 2\times 2=4 \mathrm{options} \\ \mathrm{Also} \mathrm{a} \mathrm{ball} \mathrm{can} \mathrm{be} \mathrm{given} \mathrm{to} \mathrm{same} \mathrm{person} \mathrm{twice} \mathrm{so} \mathrm{it} \mathrm{can} \mathrm{be} \mathrm{done} \mathrm{with} \\ 2 \mathrm{options} \mathrm{at} 5 \mathrm{stages} \mathrm{and} \mathrm{then} \mathrm{return} \mathrm{to} \mathrm{A} \\ \mathrm{e}.\mathrm{g}. \mathrm{B}\to \mathrm{D}\to \mathrm{B}\to \mathrm{C}\to \mathrm{D}\to \mathrm{C}\to \mathrm{A} \\ \mathrm{which} \mathrm{means} 2^5=32 \mathrm{options}, \mathrm{hence} \\ \mathrm{Total} \mathrm{ways}=3\times \left(1+4+32\right)=3\times 37=111 \end{gathered}$$

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