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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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

a + 1
na
a
0

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

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

23.

એક પરીક્ષક કોઈપણ પ્રશ્નમાં 2 થી ઓછા ગુણ આપ્યા સિવાય, 8 પ્રશ્નમાં 25 ગુણ કેટલી રીતે આપી શકે ?

$open parentheses table row bold 16 row bold 8 end table close parentheses$
$open parentheses table row bold 17 row bold 9 end table close parentheses$
$open parentheses table row bold 16 row bold 7 end table close parentheses$
₁₆P₇

24.

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 ?

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

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

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

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

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

Tips: -

open parentheses table row cell bold n bold plus bold 1 end cell row bold r end table close parentheses bold space bold equals bold space fraction numerator bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold space bold factorial over denominator bold r bold space bold factorial bold space bold left parenthesis bold n bold space bold plus bold space bold 1 bold space bold minus bold space bold r bold right parenthesis bold space bold factorial end fraction bold space bold equals bold space fraction numerator bold left parenthesis bold n bold space bold plus bold space bold 1 bold right parenthesis bold space bold n bold space bold factorial over denominator bold left parenthesis bold n bold space bold plus bold space bold 1 bold space bold minus bold space bold r bold right parenthesis bold space bold left parenthesis bold n bold space bold minus bold space bold r bold right parenthesis bold space bold factorial bold space bold r bold space bold factorial end fraction bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold space bold equals bold space fraction numerator bold n bold plus bold 1 over denominator bold n bold plus bold 1 end fraction bold space bold cross times bold space open parentheses table row bold n row bold r end table close parentheses bold therefore bold space 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 3 end table close parentheses bold space bold. bold. bold. bold space open parentheses table row cell bold n bold plus bold 1 end cell row bold n end table close parentheses bold equals bold space fraction numerator bold n bold plus bold 1 over denominator bold n end fraction open parentheses table row bold n row bold 1 end table close parentheses bold space bold cross times bold space fraction numerator bold n bold plus bold 1 over denominator bold n bold minus bold 1 end fraction bold space open parentheses table row bold n row bold 2 end table close parentheses bold space bold cross times bold space fraction numerator bold n bold plus bold 1 over denominator bold n bold minus bold 2 end fraction bold space open parentheses table row bold n row bold 3 end table close parentheses bold space bold cross times bold space bold. bold. bold. bold space bold cross times bold space fraction numerator bold n bold plus bold 1 over denominator bold 1 end fraction bold space open parentheses table row bold n row bold n end table close parentheses bold equals 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 bold space bold cross times 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

27.

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)
$fraction numerator bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold left parenthesis bold n bold plus bold 2 bold right parenthesis over denominator bold 2 end fraction$