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5 marbles and 4 identical bags

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by faraz_jeddah » Thu May 23, 2013 3:44 am
In how many ways can 5 different marbles be distributed in 4 identical pockets?

A - 24
B - 51
C - 120
D - 625
E - 1024

oa - B
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by mkdureja » Thu May 23, 2013 4:32 am
faraz_jeddah wrote:In how many ways can 5 different marbles be distributed in 4 identical pockets?

A - 24
B - 51
C - 120
D - 625
E - 1024

oa - B
Case 1: All 5 marbles in 1 pocket.
As all pockets are identical, no. of ways it can be done = 1

Case 2: 4 marbles in 1 plus 1 marble in another.
Again, all pockets are identical, we just have to find no. of ways of choosing 1 marble out of 5 marbles = 5C1 = 5

Case 3: 3 marbles + 2 marbles, No. of ways = 5C2 = 10

Case 4: 1 + 1 + 3
No. of ways = Choose 3 out of 5 to form a group, other 2 can go in any pocket as pockets are all identical.
= 5C3 = 10

Case 5: 1 + 2 + 2
Choose 1st out of 5: 5 ways
Choose 2 out of 4: 4C2/2 = 3 ways.
(Our of group of ABCD, choosing 2 has 6 ways, but in our case choosing AB or choosing CD are both same, so we have to divide it by 2.)
Total = 5 X 3 = 15.

Case 6: 1 + 1 + 1 + 2
Choose 2 of 5 = 5C2 = 10.

TOTAL: 1 + 5 + 10 + 10 + 15 + 10 =51
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by faraz_jeddah » Thu May 23, 2013 5:06 am
mkdureja wrote:
faraz_jeddah wrote:In how many ways can 5 different marbles be distributed in 4 identical pockets?

A - 24
B - 51
C - 120
D - 625
E - 1024

oa - B
Case 1: All 5 marbles in 1 pocket.
As all pockets are identical, no. of ways it can be done = 1

Case 2: 4 marbles in 1 plus 1 marble in another.
Again, all pockets are identical, we just have to find no. of ways of choosing 1 marble out of 5 marbles = 5C1 = 5

Case 3: 3 marbles + 2 marbles, No. of ways = 5C2 = 10

Case 4: 1 + 1 + 3
No. of ways = Choose 3 out of 5 to form a group, other 2 can go in any pocket as pockets are all identical.
= 5C3 = 10

Case 5: 1 + 2 + 2
Choose 1st out of 5: 5 ways
Choose 2 out of 4: 4C2/2 = 3 ways.
(Our of group of ABCD, choosing 2 has 6 ways, but in our case choosing AB or choosing CD are both same, so we have to divide it by 2.)
Total = 5 X 3 = 15.

Case 6: 1 + 1 + 1 + 2
Choose 2 of 5 = 5C2 = 10.

TOTAL: 1 + 5 + 10 + 10 + 15 + 10 =51
So since it mentions identical bags, there is no difference between choosing 4 then 1 and choosing 1 then 4.

Correct?
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