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In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical?
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In how many different ways may 12 things,4each of three varieties be d...
Problem Statement:
In how many different ways may 12 things, 4 each of three varieties, be distributed equally among 2 persons? Things of the same variety are assumed to be identical.
Solution:
To find the number of different ways to distribute the items equally among 2 persons, we can use the concept of combinations and permutations.
Step 1: Identifying the Varieties
Let's first identify the three varieties of items. Since there are 4 items each of three varieties, we can represent them as A, B, and C.
Step 2: Distributing Varieties
The first step is to distribute the three varieties among the two persons. There are three possibilities for each variety: Person 1 gets all, Person 2 gets all, or an equal distribution among both.
Case 1: Person 1 gets all varieties
In this case, Person 1 gets all the 4 items of each variety. Since the items of the same variety are assumed to be identical, there is only one way to distribute them.
Case 2: Person 2 gets all varieties
Similar to Case 1, there is only one way to distribute all the items to Person 2.
Case 3: Equal distribution among both persons
In this case, each person gets an equal number of items from all three varieties. To find the number of ways to distribute equally, we can use the concept of combinations.
Since there are 4 items of each variety, we need to distribute 4 items among 2 persons. This can be represented as distributing 4 identical items into 2 distinct boxes.
We can use the formula for combinations to find the number of ways to distribute the items equally: C(n + r - 1, r), where n is the number of items and r is the number of persons.
For this case, the number of ways to distribute the items equally among both persons is C(4 + 2 - 1, 2) = C(5, 2) = 10.
Step 3: Total Number of Ways
To find the total number of ways to distribute the items equally among 2 persons, we need to sum up the possibilities from all the cases.
Total number of ways = Number of ways in Case 1 + Number of ways in Case 2 + Number of ways in Case 3
Total number of ways = 1 + 1 + 10 = 12
Therefore, there are 12 different ways to distribute 12 identical items, 4 each of three varieties, equally among 2 persons.
In how many different ways may 12 things, 4 each of three varieties, be distributed equally among 2 persons? Things of the same variety are assumed to be identical.
Solution:
To find the number of different ways to distribute the items equally among 2 persons, we can use the concept of combinations and permutations.
Step 1: Identifying the Varieties
Let's first identify the three varieties of items. Since there are 4 items each of three varieties, we can represent them as A, B, and C.
Step 2: Distributing Varieties
The first step is to distribute the three varieties among the two persons. There are three possibilities for each variety: Person 1 gets all, Person 2 gets all, or an equal distribution among both.
Case 1: Person 1 gets all varieties
In this case, Person 1 gets all the 4 items of each variety. Since the items of the same variety are assumed to be identical, there is only one way to distribute them.
Case 2: Person 2 gets all varieties
Similar to Case 1, there is only one way to distribute all the items to Person 2.
Case 3: Equal distribution among both persons
In this case, each person gets an equal number of items from all three varieties. To find the number of ways to distribute equally, we can use the concept of combinations.
Since there are 4 items of each variety, we need to distribute 4 items among 2 persons. This can be represented as distributing 4 identical items into 2 distinct boxes.
We can use the formula for combinations to find the number of ways to distribute the items equally: C(n + r - 1, r), where n is the number of items and r is the number of persons.
For this case, the number of ways to distribute the items equally among both persons is C(4 + 2 - 1, 2) = C(5, 2) = 10.
Step 3: Total Number of Ways
To find the total number of ways to distribute the items equally among 2 persons, we need to sum up the possibilities from all the cases.
Total number of ways = Number of ways in Case 1 + Number of ways in Case 2 + Number of ways in Case 3
Total number of ways = 1 + 1 + 10 = 12
Therefore, there are 12 different ways to distribute 12 identical items, 4 each of three varieties, equally among 2 persons.
Community Answer
In how many different ways may 12 things,4each of three varieties be d...
Let the 3 varieties of 12 things be-
A, B and C
n(A) =n(B) = n(C) =4
To distribute the elements of Set A, B and C equally (quantitatively) among 3 individuals…
We first have to find the number of ways in which 12 things can be made into a group of three given the conditions…
We have 3 sets of four identical things… so we can make a group of 4 out of these 3 groups in 3X3X3X3 ways… ( every object can be chosen from any of the three sets) IF THE ORDER MATTERS
Now, that we have the groups ready we have to distribute among 3 people which can be done in 3! Ways.
So, the answer is 81X3!=486 ways
IF THE ORDER DOES NOT MATTER, THEN…
No. Of ways =3+6+9=18 ways
( all same +3 same+2 same).
Now, we can distribute them in 3!
So, 18X3!=108 ways.
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In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical?
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In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical?.
In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In how many different ways may 12 things,4each of three varieties be distributed equally among 2 persons ?things of same variety are assumed to be identical?.
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