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The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _______. (in integers)
Correct answer is '56'. Can you explain this answer?
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The number of ways, 16 identical cubes, of which 11 are blue and rest ...
First we arrange 5 red cubes in a row and assume there are x1, x2, x3, x4, x5 and x6 blue cubes between them.
Here, x1 + x2 + x3 + x4 + x5 + x6 = 11 and x2, x3, x4, x5 ≥ 2; x1,x6 ≥ 0
Let x2 = t2 + 2, x3 = t3 + 2, x4 = t4 + 2, x5 = t5 + 2, where x1,t2, t3, t4, t5,x6 ≥ 0
So x1 + t2 + t3 + t4 + t5 + x6 = 3
Required number of solutions = 8C5 = 56
Here, x1 + x2 + x3 + x4 + x5 + x6 = 11 and x2, x3, x4, x5 ≥ 2; x1,x6 ≥ 0
Let x2 = t2 + 2, x3 = t3 + 2, x4 = t4 + 2, x5 = t5 + 2, where x1,t2, t3, t4, t5,x6 ≥ 0
So x1 + t2 + t3 + t4 + t5 + x6 = 3
Required number of solutions = 8C5 = 56
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The number of ways, 16 identical cubes, of which 11 are blue and rest ...
Approach:
To solve this problem, we need to arrange the 16 identical cubes in a row such that there are at least 2 blue cubes between any two red cubes. We can break down the problem into smaller steps to find the total number of ways to arrange the cubes.
Step 1: Fix the positions of the red cubes
- Since there are 5 red cubes, we need to fix their positions in the row. The number of ways to do this is given by selecting 5 positions out of the 16 available positions, which can be calculated using combinations.
- C(16, 5) = 16! / (5! * (16-5)!) = 4368
Step 2: Arrange the blue cubes
- Now, we need to arrange the remaining 11 blue cubes in the row such that there are at least 2 blue cubes between any two red cubes.
- Since we have 5 red cubes fixed in their positions, we have 6 spaces between them where the blue cubes can be placed.
- We need to distribute 11 blue cubes into 6 spaces, which can be done using stars and bars method (combinations with repetitions).
- The number of ways to do this is given by C(11 + 6 - 1, 6 - 1) = C(16, 5) = 16! / (5! * (16-5)!) = 4368
Step 3: Calculate the total number of ways
- To find the total number of ways to arrange the cubes, we multiply the number of ways from Step 1 and Step 2.
- Total number of ways = 4368 * 4368 = 1905216
Therefore, the total number of ways to arrange the 16 identical cubes in a row such that there are at least 2 blue cubes between any two red cubes is 1905216.
To solve this problem, we need to arrange the 16 identical cubes in a row such that there are at least 2 blue cubes between any two red cubes. We can break down the problem into smaller steps to find the total number of ways to arrange the cubes.
Step 1: Fix the positions of the red cubes
- Since there are 5 red cubes, we need to fix their positions in the row. The number of ways to do this is given by selecting 5 positions out of the 16 available positions, which can be calculated using combinations.
- C(16, 5) = 16! / (5! * (16-5)!) = 4368
Step 2: Arrange the blue cubes
- Now, we need to arrange the remaining 11 blue cubes in the row such that there are at least 2 blue cubes between any two red cubes.
- Since we have 5 red cubes fixed in their positions, we have 6 spaces between them where the blue cubes can be placed.
- We need to distribute 11 blue cubes into 6 spaces, which can be done using stars and bars method (combinations with repetitions).
- The number of ways to do this is given by C(11 + 6 - 1, 6 - 1) = C(16, 5) = 16! / (5! * (16-5)!) = 4368
Step 3: Calculate the total number of ways
- To find the total number of ways to arrange the cubes, we multiply the number of ways from Step 1 and Step 2.
- Total number of ways = 4368 * 4368 = 1905216
Therefore, the total number of ways to arrange the 16 identical cubes in a row such that there are at least 2 blue cubes between any two red cubes is 1905216.
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The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is _______. (in integers)Correct answer is '56'. Can you explain this answer?
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