Examples on the Generalized Permutations and Combinations

 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 : 

$$\left(\begin{array}{c}3-1+4\\ 4\end{array}\right)=15$$ Sol 3 : 

$$\left(\begin{array}{c}7-1+5\\ 7-1\end{array}\right)=\left(\begin{array}{c}11\\ 6\end{array}\right)=462$$ Sol 4 : (1)

$$\left(\begin{array}{c}4-1+6\\ 4-1\end{array}\right)=\left(\begin{array}{c}4-1+6\\ 6\end{array}\right)=\left(\begin{array}{c}9\\ 6\end{array}\right)=84$$ (2)

$$\left(\begin{array}{c}4-1+8\\ 4-1\end{array}\right)=\left(\begin{array}{c}11\\ 3\end{array}\right)=165$$