P and C doubt

Question

In how many ways can 10 different books be distributed to three students A B C so that  student A receives at least one book?

 

In how many ways can 10 different books be distributed to three students ?

 

In how many ways can 10 different books be distributed to three students A B C so that  each student  receives at least 3 books?

How to solve these types of questions...

Any type of assistance would be appreciated..

Comments

Ans of 1 question is 58025... I m not getting this by your method....

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@rupendra u have deleted ur comment...

Answer 1

When items are of different kind (as in your question)

1) 10 different items , 3 different persons(a,b,c) 

if there is no extra condition like , A must have K books like-like then adopt following method

B₁ : book one has 3 ways to deliver , either to a or b or c = 3 ways

B₂ : 3 ways , either to a or b or c...

B₃  : 3 ways...

so B₁ to B₁₀ every book can be deliver in 3 ways so total ways 3*3*3*3...10 times =3¹⁰

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Now let suppose we have condition like A must have one book at least , so in that case first give 'a' that book , one book out of 10 can be selected in 10 ways (10C1), now remaining 9 books follow the above approach

so total ways = 10*3⁹

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Here every one must have atleast 3 books

for a it's 10C3 ways then remaining 7 give b any 3 (7C3 ways) remaining 4 give c any 3(4C3 ways) so remaining 1 can be deliver to anyone => 3 ways

total =10C3 * 7C3 * 4C3 * 3 ways

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When items are identical kinds

10 same kinds of items , among 3 persons

Case 1 )    a+b+c=10   ;     a>=1 , b>=0 , c>=0

case 2 )    a+b+c=10   ;     a,b,c>=0

case 3)     a+b+c=10   ;     a,b,c>=3

you can follow below link to know how to solve this kind of equations...

https://gateoverflow.in/158719/number-of-solutions#a158731