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MCQ

Fill in the Blanks

If the coefficients of x^{7} and x^{8} in `2 + x^n/3` are equal, then n is ______.

#### Options

56

55

45

15

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#### Solution

If the coefficients of x^{7} and x^{8} 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, T_{8} = `""^"n""C"_7 (2)^("n" - 7) (x/3)^7`

= `""^"n""C"_7 (2^("n" - 7))/3^7 x^7`

And T_{9} = `""^"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 x^{7} = coefficient x^{8})

⇒ `"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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