Ex 1 : How many strings of length 4 be formed from the English alphabet with repetition allowed ?
Ex 2 : How many ways are there to select 4 pieces with replacement of a fruit from a bowl containing apples , oranges , pears if the order in which the pieces are selected does not matter (there are at least four pieces of each type of fruit) ?
Ex 3 : How many ways are there to select five bills from a cash box containing $1 , $2 , $5 , $10 , $20 , $50 and $100 bills ?
Ex 4 : Suppose that a cookie shop has four different kinds od cookies
(1) How man different was can 6 cookies be chosen ?
(2) How man different was can 12 cookies be chosen such that at least one cookie is chosen from each type ?
Sol 1 :
\[{26^4} = 456976\]
Sol 2 :\[\left( {\begin{array}{*{20}{c}} {3 - 1 + 4}\\ 4 \end{array}} \right) = 15\] Sol 3 :
\[\left( {\begin{array}{*{20}{c}} {7 - 1 + 5}\\ {7 - 1} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {11}\\ 6 \end{array}} \right) = 462\] Sol 4 : (1)
\[\left( {\begin{array}{*{20}{c}} {4 - 1 + 6}\\ {4 - 1} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {4 - 1 + 6}\\ 6 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 9\\ 6 \end{array}} \right) = 84\] (2)
\[\left( {\begin{array}{*{20}{c}}
{4 - 1 + 8}\\
{4 - 1}
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{11}\\
3
\end{array}} \right) = 165\]