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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________

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answer is 56.

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$\begin{array}{l} x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} = 11 \\ x_{1}, x_{6} \geq 0, x_{2}, x_{3}, x_{4}, x_{5} \geq 2 \\ x_{2} = t_{1} + 2 \\ x_{3} = t_{3} + 2 \\ x_{4} = t_{4} + 2 \\ x_{5} = t_{5} + 2 \\ x_{1}, t_{2}, t_{3}, t_{4}, t_{5}, x_{6} \geq 0 \end{array}$

No. of solutions $= {}^{6 + 3 - 1}C_{3} = {}^{8}C_{3} = 56$

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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________