Question 1206918: An artist donates 3 different sculptures and 6 different vases to a museum. The exhibits will be arranged in a row. Find the number of possible arrangements if (a.) The 3 sculptures are placed together. (b.) The 3 sculptures are placed in the middle. (c.) Each sculpture must be placed between two vases.
Found 2 solutions by Edwin McCravy, mccravyedwin:
Answer by Edwin McCravy(20054)   (Show Source):
You can put this solution on YOUR website!
(a.)
SSSVVVVVV, VSSSVVVVV, VVSSSVVVV, VVVSSSVVV, VVVVSSSVV, VVVVVSSSV, VVVVVVSSS
As you see above, there are 7 ways to put 0,1,...,6 V's before the 3 S's.
For each of those 7 ways, there are 6! ways to arrange the V's and 3! ways to
arrange the S's. That's (7)(6!)(3!) = 30240 ways.
(b.) That's just 1/7 of the above, the case VVVSSSVVV, (6!)(3!) = 4320.
(c.) V_V_V_V_V_V
We can arrange the V's in 5! ways. Label the sculptures S1, S2, S3
For each of those 5! ways, we choose
1. a blank to place S1 in 5 ways.
2. a blank to place S2 in 4 ways.
3. a blank to place S3 in 3 ways.
That's (6!)(5)(4)(3) = 43200 ways.
Edwin
Answer by mccravyedwin(406)  (Show Source):
You can put this solution on YOUR website!
This is just a correction of the above. I wrote 5! twice when it should have been 6!
But the answer is the same.
(a.)
SSSVVVVVV, VSSSVVVVV, VVSSSVVVV, VVVSSSVVV, VVVVSSSVV, VVVVVSSSV, VVVVVVSSS
As you see above, there are 7 ways to put 0,1,...,6 V's before the 3 S's.
For each of those 7 ways, there are 6! ways to arrange the V's and 3! ways to
arrange the S's. That's (7)(6!)(3!) = 30240 ways.
(b.) That's just 1/7 of the above, the case VVVSSSVVV, (6!)(3!) = 4320.
(c.) V_V_V_V_V_V
We can arrange the V's in 6! ways. Label the sculptures S1, S2, S3
For each of those 6! ways, we choose
1. a blank to place S1 in 5 ways.
2. a blank to place S2 in 4 ways.
3. a blank to place S3 in 3 ways.
That's (6!)(5)(4)(3) = 43200 ways.
Edwin
|
|
|
