Question

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.

Answer 1

Answer:

Therefore there are 7 ways to distribute the marbles.

Step-by-step explanation:

There are 2n red marbles, 2n green marbles and 2n blue marbles.

For each person to receive 3n marbles the following are the ways in which the distribution can be made. Each person can receive:

i) 1n red marbles,  1n blue marbles, 1n green marbles

ii) 2n red marbles, 1n green marbles

iii) 2n red marbles, 1n blue marbles

iv) 2n green marbles, 1n red marbles

v) 2n green marbles, 1n blue marbles

vi) 2n blue marbles, 1n red marbles

vii) 2n blue marbles, 1n green marbles

Therefore there are 7 ways to distribute the marbles.

Answer 2

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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