Using five basic colours (Red, Blue, Green, Black, Pink), different colours are made by mixing these two or more colours in equal proportion. n vertical stripes are painted by all colours available such that no two consecutive stripes have same colour. Then number of ways of painting ‘n’ stripes such that only alternate stripes are painted by a basic colour is

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$2 {(130)}^{n / 2},$ if n is odd

$31 {(130)}^{\frac{n - 1}{2}},$ if n is even

$31 {(130)}^{\frac{n - 1}{2}},$ if n is odd

$2 {(130)}^{n / 2},$ if n is even

answer is C, D.

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Detailed Solution

No. of basic colours = 5
No. of colours formed by mixing = $^{5} C_{2} +^{5} C_{3} +^{5} C_{4} +^{5} C_{5} = 26$
If n is even
No. of ways: $5^{n / 2} . {(26)}^{n / 2} + 5^{n / 2} . {(26)}^{n / 2} = 2 {(130)}^{n / 2}$
If n is odd
No. of : $5^{\frac{n + 1}{2}} . {(26)}^{\frac{n + 1}{2}} + 5^{\frac{n - 1}{2}} . {(26)}^{\frac{n - 1}{2}} = 31 {(130)}^{\frac{n - 1}{2}}$