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Find the 6th term of the expansion (y1/2 + x1/3)n , if the binomial coefficient of the 3rd term from the end is 45.

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Given : 3rd term from the end =45 

To Find : 6th term 

For (y1/2 + x1/3) n

a = y1/2, b = x1/3

We have, tr+1 = (nr)an-rbr

As n = n, therefore there will be total (n+1) terms in the expansion. 

3rd term from the end = (n+1-3+1)th i.e. (n-1)th term from the starting

For (n-1)th term, r = (n-1-1) = (n-2)

t(n-1) = t(n-2)+1

Therefore 3rd term from the end = \(\frac{n(n-1)}{2}\) (Y)(X)\(\frac{n-2}{3}\)

Therefore coefficient 3rd term from the end = \(\frac{n(n-1)}{2}\) 

45 =   \(\frac{n(n-1)}{2}\) 

•90 = n (n-1) 

• 10 (9) = n (n-1) 

Comparing both sides, n = 10 

For 6th term, r = 5

t6 = t5+1

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