# Find the sixth term of the expansion n(y12+x13)n, if the binomial coefficient of the third term from the end is 45. - Mathematics

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Find the sixth term of the expansion (y^(1/2) + x^(1/3))^"n", if the binomial coefficient of the third term from the end is 45.

#### Solution

The given expression is (y^(1/2) + x^(1/3))^"n"

Since the binomial coefficient of third term from the end = Binomial coefficient of third term from the beginning = nC2

nC2 = 45

⇒ ("n"("n" - 1))/2 = 45

⇒ n2 – n = 90

⇒ n2 – n – 90 = 0

⇒ n2 – 10n + 9n – 90 = 0

⇒ n(n – 10) + 9(n – 10) = 0

⇒ (n – 10)(n + 9) = 0

⇒ n = 10, n = –9

⇒ n = 10, n ≠ – 9

So, the given expression becomes (y^(1/2) + x^(1/3))^10

Sixth term is this expression T6 = T5+1

= ""^10"C"_5 (y^(1/2))^(10 - 5)  (x^(1/3))^5

= ""^10"C"_5  y^(5/2) * x^(5/3)

= 252  y^(5/2) x^(5/3)

Hence, the required term = 252  y^(5/2) * x^(5/3)

Concept: Binomial Theorem for Positive Integral Indices
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Exercise | Q 8 | Page 143
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