Answer to Question #270613 in Discrete Mathematics for farooq

Question

A toy shop has 15 airplanes, 15 buses, 17 trains, and 20 bikes in the stock. a. How many ways are there for a person to take 15 toys home if all the airplanes are identical, all the buses are identical, all the trains are identical and all the bikes are identical? b. How many ways are there for a person to take 15 toys home if all the airplanes are distinct, all the buses are distinct, all the trains are distinct and all the bikes are distinct? c. How many ways are there for a person to take 25 toys home if all the airplanes are identical, all the buses are identical, all the trains are identical and all the bikes are identical?  

Answer 1

a.

it's same to number of ways to place n(15) balls into m(4) boxes:

$n=C^{m+n-1}_n=C^{15}_{15+4-1}=\frac{18!}{15!3!}=816$

b.

it' s number of ways to take 15 items from total 67:

$n=C^{15}_{67}=\frac{67!}{15!52!}=345780890878896$

c.

$n=C^{m+n-1}_n=C^{25}_{25+4-1}=\frac{28!}{25!3!}=3276$