# If the coefficients of x7 and x8 in 2+xn3 are equal, then n is ______. - Mathematics

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MCQ
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If the coefficients of x7 and x8 in 2 + x^n/3 are equal, then n is ______.

#### Options

• 56

• 55

• 45

• 15

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#### Solution

If the coefficients of x7 and x8 in 2 + x^"n"/3 are equal, then n is 55.

Explanation:

Since "T"_("r" + 1) = ""^"n""C"_"r"  "a"^("n" - "r")  x^"r" in expansion of (a + x)n

Therefore, T8 = ""^"n""C"_7 (2)^("n" - 7)  (x/3)^7

= ""^"n""C"_7  (2^("n" - 7))/3^7  x^7

And T9 = ""^"n""C"_8  (2)^("n" - 8)  (x/3)^8

= ""^"n""C"_8  (2^("n" - 8))/3^8  x^8

Therefore, ""^"n""C"_7  (2^("n" - 7))/3^7

= ""^"n""C"_8 (2^("n" - 8))/3^8   ....(Since it is given that coefficient of x7 = coefficient x8)

⇒ "n"/((7)("n" - 7)) xx (8("n" - 8))/"n" = (2^("n" - 8))/3^8 * 3^7/(2^("n" - 7))

⇒ 8/("n" - 7) = 1/6

⇒ n = 55

Concept: Binomial Theorem for Positive Integral Indices
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Chapter 8: Binomial Theorem - Solved Examples [Page 139]

#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 8 Binomial Theorem
Solved Examples | Q 17 | Page 139
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