JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Permutations and Combinations

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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 ____.

Answer: 56

Solution

We have 11 blue cubes and 5 red cubes. Let the red cubes divide the row into 6 segments (before the first red, between consecutive reds, and after the last red).

Let

x_{1}, x_{2}, \dots, x_{6}

be the number of blue cubes in each segment. We need:

x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} = 11

with

x_{1}, x_{6} \geq 0

and

x_{2}, x_{3}, x_{4}, x_{5} \geq 2

(at least 2 blue cubes between any two red cubes).

Substitute

x_{i} = t_{i} + 2

for

i = 2, 3, 4, 5

x_{1} + (t_{2} + 2) + (t_{3} + 2) + (t_{4} + 2) + (t_{5} + 2) + x_{6} = 11

x_{1} + t_{2} + t_{3} + t_{4} + t_{5} + x_{6} = 3

where all variables

\geq 0

.

Number of non-negative integer solutions:

(6 - 1 3 + 6 - 1) = (5 8) = (3 8) = 56

JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Integer answer

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JEE Main · 2022 · 27 Jun 2022 · Shift 1 · Permutations and Combinations

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