(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
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