Question

Find n, if $\rm ^{18}C_{2n} =\ \rm ^{18}C_{n^{2}+3}$

Answer 1

Solution:

Given,

$\rm ^{18}C_{2n} =\ \rm ^{18}C_{n^{2}+3}$

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We know that ,

If $\rm ^{n}C_{r} =\ \rm ^{n}C_{n^{p}$

then either r = p or n = r + p

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Here ,

n = 18 , r = 2n , p = n² + 3

r + p = n

=> 2n + n² + 3 = 18

=> n² + 2n - 15 = 0

Splitting the middle term , we get,

=> n² + 5n - 3n - 15 = 0

=> n( n + 5 ) - 3( n + 5 ) = 0

=> ( n + 5 )( n - 3 ) = 0

=> n + 5 = 0 or n - 3 = 0

=> n = -5 or n = 3

Therefore ,

n = 3

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