Let n be a positive integer. 2n blue marbles, 2n red marbles, and 2n green marbles are given to two people. Marbles of the same colour are considered indistinguishable. In how many ways can the marbles be distributed so that each person gets 3n marbles? Justify your answer fully.
Please explain each step if possible.
Answers
Given : n be a positive integer. 2n blue marbles, 2n red marbles, and 2n green marbles are given to two people. balls identical except color
To find : how many ways can the marbles be distributed so that each person gets 3n marbles
Step-by-step explanation:
Blue Marbles = 2n
Red Marbles = 2n
Green Marbles = 2n
Given 3n to one person
Any marble can be from 0 to 2n
if Blue Marbles = 0
then Red Marbles can be from n to 2n and corresponding Green marbles 2n to n ( number of ways = n + 1)
if Blue Marbles = 1
then Red Marbles can be from n-1 to 2n and corresponding Green marbles 2n to n-1 ( number of ways = n+2)
and so on
Blue Marbles Red + green Marbles Number of Ways
0 3n n+ 1
1 3n-1 n+2
2 3n-1 n+ 3
..
--
n 2n 2n+1
n+1 2n-1 2n
n+2 2n-2 2n-1
--
2n-1 n+1 n+2
2n n n+1
Total number of ways
= (n + 1) + ( n + 2) +.............2n.+ (2n+1) + 2n +...............+(n+2) + (n+1)
= 2( (n + 1) + ( n + 2) +.............2n) + 2n+1
= 2 (n/2) ( n+1 + 2n) + 2n + 1
= n(3n + 1) + 2n + 1
= 3n² + n + 2n + 1
= 3n² + 3n + 1
3n² + 3n + 1 : are the number of ways marbles can be distributed so that each person gets 3n marbles
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