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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

21.

અસંખ્ય લાલ, સફેદ, કાળા અને લીલા દડાઓમાંથી દરેક રંગના દડા આવે તે રીતે 11 દડા પસંદ કરવાના પ્રકાર કેટલા થાય ?

165
330
120
અસંખ્ય

22. જો (1 + x)¹⁰⁰ = c₀ + c₁x + c_zx² + ... + c₁₀₀x¹⁰⁰ તો $bold 2 to the power of bold 1 bold space bold c subscript bold 0 over bold 1 bold space bold plus bold space bold 2 to the power of bold 2 bold space bold c subscript bold 1 over bold 2 bold space bold plus bold space bold 2 to the power of bold 2 bold space bold c subscript bold 2 over bold 3 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold 2 to the power of bold 101 bold space bold c subscript bold 100 over bold 101 bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space$

$bold 3 to the power of bold 101 over bold 101$
$fraction numerator bold 3 to the power of bold 101 bold plus bold 1 over denominator bold 101 end fraction$
$fraction numerator bold 3 to the power of bold 101 bold minus bold 1 over denominator bold 101 end fraction$
$fraction numerator bold 2 to the power of bold 101 bold minus bold 1 over denominator bold 101 end fraction$

23.

એક વિદ્યાર્થી ત્રણ વિષયની પરીક્ષા આપે છે. દરેક વિષયના પેપર મહત્તમ n ગુણના છે. તે 2n ગુણ કેટલી રીતે મેળવી શકે ?

$fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 2 end fraction$
$open parentheses table row cell bold 3 bold n end cell row cell bold 2 bold n end cell end table close parentheses$
n(n+1)(n+2)
$fraction numerator bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold left parenthesis bold n bold plus bold 2 bold right parenthesis over denominator bold 2 end fraction$

24. જો $open parentheses table row cell bold n bold plus bold 1 end cell row bold 1 end table close parentheses bold space open parentheses table row cell bold n bold plus bold 1 end cell row bold 2 end table close parentheses bold space open parentheses table row cell bold n bold plus bold 1 end cell row bold n end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold space bold. bold space$

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

25.

ગણિત, ભૌતિક અને રસાયણ વિષયનાં પુસ્તકોના સમૂહમાં દરેક વિષયનાં ઓછામાં ઓછા 10 પુસ્તકો (ધોરણ-12ના) છે. આ સમુહમાંથી 5 પુસ્તકો કેટલી રીતે પસંદ કરી શકાય ?

26. જો (1 + x)ⁿ = c₀ + c₁x + c₂x² + ... + c_nxⁿ તો c₀ - $bold 1 over bold 2 bold space bold c subscript bold 1 bold plus bold space bold 1 over bold 3 bold space bold c subscript bold 2 bold space bold minus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold c subscript bold n bold space bold equals bold space bold. bold. bold. bold. bold. bold space$

$fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction$
$fraction numerator bold 2 to the power of bold n bold plus bold 1 end exponent bold minus bold 1 over denominator bold n bold plus bold 1 end fraction$
$fraction numerator bold 2 to the power of bold n bold plus bold 1 end exponent bold plus bold 1 over denominator bold n bold plus bold 1 end fraction$
$fraction numerator bold 2 over denominator bold n bold plus bold 1 end fraction$

fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction

Tips: -

bold c subscript bold 0 bold minus bold 1 over bold 2 bold space bold c subscript bold 1 bold space bold plus bold space bold 1 over bold 3 bold space bold c subscript bold 2 bold space bold minus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold c blank with blank below bold equals bold space open parentheses table row bold n row bold 0 end table close parentheses bold space bold minus bold space bold 1 over bold 2 bold space open parentheses table row bold n row bold 1 end table close parentheses bold space bold plus bold space bold 1 over bold 3 bold space open parentheses table row bold n row bold 2 end table close parentheses bold space bold minus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space open parentheses table row bold n row bold n end table close parentheses bold equals bold space bold 1 bold space bold minus bold space bold 1 over bold 2 bold space bold n bold space bold plus bold space fraction numerator bold n bold left parenthesis bold n bold minus bold 1 bold right parenthesis over denominator bold 6 end fraction bold space bold minus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold left square bracket bold space bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold space bold minus bold space fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 2 end fraction bold space bold plus bold space fraction numerator bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold space bold n bold left parenthesis bold n bold minus bold 1 bold right parenthesis over denominator bold 6 end fraction bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space bold right square bracket bold space bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space open square brackets open parentheses table row cell bold n bold plus bold 1 end cell row bold 1 end table close parentheses bold minus open parentheses table row cell bold n bold plus bold 1 end cell row bold 2 end table close parentheses bold plus open parentheses table row cell bold n bold plus bold 1 end cell row bold 3 end table close parentheses bold minus bold. bold. bold. bold plus bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold space open parentheses table row cell bold n bold plus bold 1 end cell row cell bold n bold plus bold 1 end cell end table close parentheses close square brackets bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space open square brackets bold 1 bold minus open curly brackets open parentheses table row cell bold n bold plus bold 1 end cell row bold 0 end table close parentheses bold minus open parentheses table row cell bold n bold plus bold 1 end cell row bold 1 end table close parentheses bold plus open parentheses table row cell bold n bold plus bold 1 end cell row bold 2 end table close parentheses bold minus bold space bold. bold. bold. bold space bold plus bold space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold n bold plus bold 1 end exponent bold space open parentheses table row cell bold n bold plus bold 1 end cell row cell bold n bold plus bold 1 end cell end table close parentheses close curly brackets close square brackets bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold left square bracket bold space bold 1 bold space bold minus bold space bold left parenthesis bold 1 bold space bold minus bold space bold 1 bold right parenthesis to the power of bold n bold plus bold 1 end exponent bold right square bracket bold space bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold left square bracket bold space bold 1 bold space bold minus bold space bold 0 bold right square bracket bold space bold equals bold space fraction numerator bold 1 over denominator bold n bold plus bold 1 end fraction

27. જો $bold sum from bold r bold equals bold 0 to bold n of bold thin space bold left parenthesis bold minus bold 1 bold right parenthesis to the power of bold r bold space bold left parenthesis bold a bold minus bold r bold right parenthesis bold space open parentheses table row bold n row bold r end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold space bold.$