(Q). Distribute 8 distinct objects into 3 distinct groups such that each group has at least 2 objects.
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(Q.) We have to distribute n distinct objects among n children such that exactly one child get no onject.
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(Q). In how many ways 8 toys can be distributed among three brothers such that such that each one receive at least one toy & no two get same number of toys & the youngest one receives the maximum number of toys.
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(Q). In how many ways 10 distinct toys can be distributed among three brothers such that each one receive at least one toy & no two get same number of toys & the youngest one receives the maximum number of toys.
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(Q). Distribute 5 different fruits among 3 bucket such that each bucket has at least one fruit.
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(Q). A teacher has 31 students in his class. one day he decided to take all students for a picnic. So he called 3 cars & 4 auto rickshaws. He himself will go with his bike. In how many ways he can make them sit in vehicle, if seating capacity of car is 5 & that of auto rickshaw is 4. Internal arrangement within vehicle does not matter.
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(Q). In the previous Example if 3 particular students insist on going by the same car then find the number of ways.
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(Q). Distribute 5 different books among 3 students such that each student gets at least one book, also order of the books matters which a student has.
Solution:-
⇒ Actually in this question value of distinct object (5) & distinct groups (3) is less. So we can do it manually but if these values becomes more then it will become very difficult to do it manually. So in that case we will be required a formula to solve this type of problem quickly.
In Distribution of distinct object in distinct groups if arrangemet of objects within group is also important then:-
Number of ways to arrange n distinct object into r distinct group is:
= n!.ⁿ ⁺ ʳ ⁻ ¹Cᵣ ₋ ₁ if blank group are allowed
= n!ⁿ ⁻ ¹Cᵣ ₋ ₁ if blank group are not allowed i.e. each group must have at least one object
(Q). In the previous question:-
n = 5 r = 3
(i) when empty group not allowed
= 5!.⁵⁻¹C₃₋₁ = 5!.⁴C₂ = 5!.6 = 720 Answer
(ii) when empty group alloweed
= 5!.⁵⁺³⁻¹C₃₋₁ = 5!.⁷C₂ = 120×21 = 2520 Answer
(Q) In how many ways 10 different rings can be worn on 5 fingers if arrangement of rings in a finger matters.
(i) Any Finger has any number of rings.
(ii) Each Finger should have at least one ring.
Solution:-
n = 10
r = 5
(i) 10!.¹⁰⁺⁵⁻¹C₅₋₁ = 10!.¹⁴C₄ = 10!. =
Answer
(ii) 10!.¹⁰⁻¹C₅₋₁ = 10!.⁹C₄ Answer
(Q). In how many ways 5 distinct books can be distributed among 10 students such that each student gets at most 1 book.
Solution:-
(Q). In how many ways 5 distinct books can be distributed among 10 students such that each student gets at least 1 book.
Solution:-
Since number of books (5) is less than number of students (10) & each student should get at least one book. So there is no such way.
Hence 0 Answer