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દ્વિપદી પ્રમેય

(1 + x)^-n (n ∈ N) ના વિસ્તરણમાં x^r નો સહગુણક ....... છે. ( |x|<1)

$bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row cell bold n bold plus bold 1 end cell end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row cell bold r bold minus bold 1 end cell end table close parentheses$

bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses

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(1 + x)^-n ના વિસ્તરણમાં વ્યાપક પદ

$bold T subscript bold r bold plus bold 1 end subscript bold space bold equals bold space fraction numerator bold left parenthesis bold minus bold n bold right parenthesis bold space bold left parenthesis bold minus bold n bold minus bold 1 bold right parenthesis bold space bold left parenthesis bold minus bold n bold minus bold 2 bold right parenthesis bold. bold. bold. bold left parenthesis bold minus bold n bold minus bold r bold plus bold 1 bold right parenthesis over denominator bold r bold factorial end fraction bold x to the power of bold r$

$bold equals bold space fraction numerator bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space bold n bold space bold left parenthesis bold n bold space bold plus bold space bold 1 bold right parenthesis bold space bold left parenthesis bold n bold space bold plus bold space bold 2 bold right parenthesis bold space bold. bold. bold. bold space bold left parenthesis bold n bold space bold plus bold space bold r bold space bold minus bold space bold 1 bold right parenthesis over denominator bold r bold space bold factorial end fraction bold space bold x to the power of bold r bold equals bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses bold space bold x to the power of bold r$

∴ x^r નો સહગુણક $bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$

(1 - x)^-n (n ∈ N)ના વિસ્તરણમાં x^r નો સહગુણક ......... છે. ( |x| < 1)

$open parentheses table row cell bold n bold minus bold r bold plus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row bold r end table close parentheses$
$open parentheses table row cell bold n bold plus bold r bold minus bold 1 end cell row straight n end table close parentheses$
$bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space open parentheses table row cell bold n bold minus bold r bold plus bold 1 end cell row bold r end table close parentheses$

(1 + 2x)^-7 ના વિસ્તરણમાં x¹⁰ નો સહગુણક ....... છે.

$open parentheses table row bold 16 row bold 6 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 17 row bold 10 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 16 row bold 10 end table close parentheses$
$bold 2 to the power of bold 10 bold space open parentheses table row bold 16 row bold 9 end table close parentheses$

જો (1 - x + x²)^-3 = a₀ + a₁x + a₂x² + ... તો $bold sum from bold r bold equals bold 0 to bold 6 of bold space bold a subscript bold r bold space bold equals bold space bold. bold. bold. bold. bold space bold left parenthesis bold vertical line bold x bold vertical line bold space bold less than bold space bold 1 bold right parenthesis bold space$

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-7
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(a + 2b)³ (2a - b)⁴ ના વિસ્તરણમાં a³b⁴ નો સહગુણક ........ છે.

593
-16
-15
15

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