Question

How many different ways are there to distribute 7 similar flowers to

3 different people. Explain your answer.

Answer 1

solution:

Given that

No.of identical flowers (n) = 7

No.of people = 3

Since flowers are identical , we can form 4 different cases for the arrangement of the no.of flowers

into the distinct boxes

( 1 - 1 - 5)

( 1 - 2 - 4)

( 1 - 3 - 3)

( 2 - 2 - 3)

----->  ( 1 - 1 - 5) means when 2 people get 1 flower each and 3rd one gets 5 flowers.

Here,The no.of possible ways to distribute 7 flowers to 3 different people = 3! / 2! = 3 ways

( 1 , 1 , 5) : 1st one gets 1 flower , 2nd one gets 1 flower , 3rd one gets 5 flowers

( 1 , 5 , 1) : 1st one gets 1 flower , 2nd one gets 5 flowers, 3rd one gets 1 flower

( 5 , 1 , 1)   : 1st one gets 5 flowers , 2nd one gets 1 flower , 3rd one gets 1 flower

----->  ( 1 - 2 - 4) means when one people gets 1 flower ,second one gets 2 and 3rd one gets 4 flowers.

Here,The no.of possible ways to distribute 7 flowers to 3 different people = 3! = 6 ways

( 1 , 2 , 4)   : 1st one gets 1 flower , 2nd one gets 2 flowers , 3rd one gets 4 flowers

( 1 , 4 , 2)    : 1st one gets 1 flower , 2nd one gets 4 flowers , 3rd one gets 2 flowers

( 2 , 1 , 4)    : 1st one gets 2 flowers , 2nd one gets 1 flower , 3rd one gets 4 flowers

(2 , 4 , 1)   : 1st one gets 2 flowers , 2nd one gets 4 flower , 3rd one gets 1 flower

(4 , 1 , 2)    : 1st one gets 4 flowers , 2nd one gets 1 flower , 3rd one gets 2 flowers

(4 , 2 , 1)    : 1st one gets 4 flowers , 2nd one gets 2 flower , 3rd one gets 1 flower

----->  ( 1 - 3 - 3) means when 2 people get 3 flower each and other one gets 1 flower.

Here,The no.of possible ways to distribute 7 flowers to 3 different people = 3! / 2! = 3 ways

( 1 , 3 , 3)    : 1st one gets 1 flower , 2nd one gets 3 flowers , 3rd one gets 3 flowers

( 3 , 1 , 3)   : 1st one gets 3 flowers , 2nd one gets 1 flower , 3rd one gets 3 flowers

( 3 , 3 , 1)   : 1st one gets 3 flowers , 2nd one gets 3 flowers , 3rd one gets 1 flower

----->  ( 2 - 2 - 3) means when 2 people get 2 flower each and 3rd one gets 3 flowers.

Here,The no.of possible ways to distribute 7 flowers to 3 different people = 3! / 2! = 3 ways

( 2 , 2 , 3)   : 1st one gets 2 flowers , 2nd one gets 2 flowers , 3rd one gets 3 flowers

( 2 , 3 , 2)    : 1st one gets 2 flowers , 2nd one gets 3 flowers , 3rd one gets 2 flowers

( 3 , 2 , 2)    : 1st one gets 3 flowers , 2nd one gets 2 flowers , 3rd one gets 2 flowers

Hence,Total No.of ways to distribute 7 flowers to 3 distinct people = 3+6+3+3 = 15

[Note: The no.of compositions of N into K parts is given by (N-1)C(K-1)

Here, N = 7 , K = 3  

Therefore,Total No.of ways to distribute 7 flowers to 3 distinct people = 6C2 = 15 ways ]