Combinations

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Question

Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour):

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Solution

$G G G G G G G / B B B B B B B B$
${One gap out of nine can be taken in}^{9} C_{1} ways and 2 green bottles can be placed.$
Now, green remaining 5
Gaps remaining 8
$5 gaps out of 8 gaps for remaining 5 {green bottles can be selected in}^{8} C_{5} .$
$\begin{array}{l} Hence, total ways =^{9} {C_{1}}^{8} C_{5} = 9 \cdot \frac{8.7.6}{1.2.3} \\ = 72.7 = 504 \end{array}$

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Combinations

Remember concepts with our Masterclasses.

Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour):

G G G G G G G/B B B B B B B B One gap out of nine can be taken in 9C1 ways and 2 green bottles can be placed. Now, green remaining 5Gaps remaining 85 gaps out of 8 gaps for remaining 5 green bottles can be selected in 8C5 . Hence, total ways =9C18C5=9⋅8.7.61.2.3=72.7=504

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