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Property 1

If fx= 1 + ...

Question

If $f (x) = (1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{15} x^{15},$ then $f (2) =$

A

1^{15}

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B

3^{15}

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C

2^{15}

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D

None of these

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Solution

The correct option is B $3^{15}$
$f (x) = (1 + x)^{15} f (2) = (1 + 2)^{15} = 3^{15}$

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(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{15} x^{15},

C_{2} + 2 C_{3} + 3 C_{4} + . . . . . . + 14 C_{15} =

(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . + C_{15} x^{15},

C_{2} + 2 C_{3} + 3 C_{4} + . . . . . . + 14 C_{15} =

If $(1 + x)^{15}$ = $C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . . . + C_{15} x^{15}$ , then

$C_{2} + 2 C_{3} + 3 C_{4} + . . . . . . . . + 14 C_{15}$ =

(1 + x)^{15} = C_{0} + C_{1} x + C_{2} x^{2} + \dots \dots + C_{15} x^{15},

find the value of

C_{2} + 2 C_{3} + 3 C_{4} + \dots \dots + 14 C_{15}

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + \dots + C_{n} x^{n},

n \sum r = 0 n \sum s = 0 (r + s) C_{r} C_{s}

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