Number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so

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6.Permutation and Combination

hard

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

A

$56$

B

$66$

C

$76$

D

$86$

(JEE MAIN-2022)

Solution

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

$x_{1}, x_{6} \geq 0, \quad x_{2}, x_{3}, x_{4}, x_{5} \geq 2$

$x_{2}=t_{1}+2$

$x_{3}=t_{3}+2$

$x_{4}=t_{4}+2$

$x_{5}=t_{5}+2$

$x_{1}, t_{2}, t_{3}, t_{4}, t_{5}, x_{6} \geq 0$

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

Standard 11

Mathematics

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