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04 Jul 2026, 21:49
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
53% (02:31) correct
47%
(03:22)
wrong
based on 85
sessions
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Not Attempted Yet
A cruise show will consist of three different 15-minute dance numbers, three different 10-minute comedic monologues, and three different 5-minute songs. In how many different orders can the nine performances be scheduled so that every block of three consecutive performances is 30 minutes long?
A. 216
B. 648
C. 1296
D. 1680
E. 2592
A. 216
B. 648
C. 1296
D. 1680
E. 2592
04 Jul 2026, 23:46
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Fun question!
How can we get a block of 3 consecutive performances such that the total block time is 30 min?
- Can we have 2 songs in a block of 3? No.
(5 + 5 = 10; we would need the last performance to be 20 min. No such option exists.)
- Can we have 2 comedic monologues in a block of 3? No.
(10+10 = 20 min; while another comedic monologue can give us 30 min in this block, it is impossible for any other block of 3 to have a length of 30 min.)
- Can we have 2 dance numbers in a block of 3? Again, No.
(15+15 = 30 min; we hit the 30 min mark with only 2 performances. The next performance will take the block length above 30 min.)
Key insight: Every block of 3 consecutive performances will have exactly 1 dance number, 1 comedy monologue, and 1 song.
What does this look like?
Example -> Say, the first block is Dance-Song-Comedy. Then, the first block looks like this: (15-5-10)
The overall sequence will look like this: 15-5-10---15-5-10---15-5-10
Observe -- Repeating the same sequence as Block 1 (Dance-Song-Comedy, in this case) is the ONLY way to ensure that every consecutive block of three has a 15, a 10, a 5.
For instance -> 15-10-5--10-5-15--15-5-10. We can see that when we use a different sequence for block 2, we instantly break the "30 min for any consecutive block of 3" condition.
Key Insight: The sequence of the first block IS the sequence for the other two blocks too.
So, every block of 3 contains 1D, 1C, and 1S. And the sequence of D, S, and C, in the first block simply repeats for the second and third blocks.
Let's now find the number of different orders of arranging the 9 performances such that every block of three consecutive performances is 30 minutes long
1) #ways of selecting 1D, 1C, and 1S for Block 1 = 3C1 x 3C1 x 3C1
2) Once selected, #ways to sequence the first block = 3!
3) #ways of selecting 1D, 1C, and 1S for Block 2 = 2C1 x 2C1 x 2C1
4) Once selected, #ways to sequence the first block = 1 (Because we must follow the same sequence as Block 1)
5) #ways of selecting 1D, 1C, and 1S for Block 3 = 1C1 x 1C1 x 1C1
6) Once selected, #ways to sequence the first block = 1 (Because we must follow the same sequence as Block 1)
Answer: 3 x 3 x 3 x 6 x 2 x 2 x 2 = 6 x 6 x 6 x 6 = 216 x 6 = 1296. Choice C.
---
Harsha
How can we get a block of 3 consecutive performances such that the total block time is 30 min?
- Can we have 2 songs in a block of 3? No.
(5 + 5 = 10; we would need the last performance to be 20 min. No such option exists.)
- Can we have 2 comedic monologues in a block of 3? No.
(10+10 = 20 min; while another comedic monologue can give us 30 min in this block, it is impossible for any other block of 3 to have a length of 30 min.)
- Can we have 2 dance numbers in a block of 3? Again, No.
(15+15 = 30 min; we hit the 30 min mark with only 2 performances. The next performance will take the block length above 30 min.)
Key insight: Every block of 3 consecutive performances will have exactly 1 dance number, 1 comedy monologue, and 1 song.
What does this look like?
Example -> Say, the first block is Dance-Song-Comedy. Then, the first block looks like this: (15-5-10)
The overall sequence will look like this: 15-5-10---15-5-10---15-5-10
Observe -- Repeating the same sequence as Block 1 (Dance-Song-Comedy, in this case) is the ONLY way to ensure that every consecutive block of three has a 15, a 10, a 5.
For instance -> 15-10-5--10-5-15--15-5-10. We can see that when we use a different sequence for block 2, we instantly break the "30 min for any consecutive block of 3" condition.
Key Insight: The sequence of the first block IS the sequence for the other two blocks too.
So, every block of 3 contains 1D, 1C, and 1S. And the sequence of D, S, and C, in the first block simply repeats for the second and third blocks.
Let's now find the number of different orders of arranging the 9 performances such that every block of three consecutive performances is 30 minutes long
1) #ways of selecting 1D, 1C, and 1S for Block 1 = 3C1 x 3C1 x 3C1
2) Once selected, #ways to sequence the first block = 3!
3) #ways of selecting 1D, 1C, and 1S for Block 2 = 2C1 x 2C1 x 2C1
4) Once selected, #ways to sequence the first block = 1 (Because we must follow the same sequence as Block 1)
5) #ways of selecting 1D, 1C, and 1S for Block 3 = 1C1 x 1C1 x 1C1
6) Once selected, #ways to sequence the first block = 1 (Because we must follow the same sequence as Block 1)
Answer: 3 x 3 x 3 x 6 x 2 x 2 x 2 = 6 x 6 x 6 x 6 = 216 x 6 = 1296. Choice C.
---
Harsha
