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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

31. અસંખ્ય લાલ, પીળા અને વાદળી રંગના દડામાંથી 12 દડા પસંદ કરવાના પ્રકાર ....... છે.

55
91
105
અસંખ્ય

32.

સમીકરણ x + y + z = 20, x ≥ 1, y ≥ 2, z ≥ 3, x, y, z ∈ Wના ઉકેલોની સંખ્યા ...... છે.

124
136
210
120

33. $bold 3 over bold 4 bold plus open parentheses bold 3 over bold 4 bold cross times bold 5 over bold 8 close parentheses bold space bold plus bold space open parentheses bold 3 over bold 4 bold cross times bold 5 over bold 8 bold cross times bold 7 over bold 12 close parentheses bold space bold plus bold space bold. bold. bold. bold space$ અનંત પદ= .......

$bold 2 bold space bold times bold space square root of bold 2 bold space bold plus bold space bold 1$
$square root of bold 2 bold space bold minus bold space bold 1$
$bold 2 bold space bold times bold space square root of bold 2 bold space bold minus bold space bold 1$
$bold 2 bold space bold times bold space square root of bold 2$

34.

15 સમાન વસ્તુઓને ત્રણ પેટીમાં કુલ કેટલી રીતે મૂકી શકાય ? (દરેક પેટીમાં વધુમાં વધુ 15 વતુ આવી શકે.)

136
3¹⁵
₁₅P₃
91

35. જો $bold c subscript bold r bold space bold equals open parentheses table row bold 30 row bold r end table close parentheses bold space bold ત ો bold space open parentheses bold c subscript bold 1 over bold c subscript bold 0 close parentheses to the power of bold 2 bold space bold plus bold space bold 8 bold space open parentheses bold c subscript bold 2 over bold c subscript bold 1 close parentheses to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space bold 30 to the power of bold 3 bold space open parentheses bold c subscript bold 30 over bold c subscript bold 29 close parentheses to the power of bold 2 bold space bold equals bold space bold. bold. bold. bold. bold space bold. bold space$

75880
76080
76880
76808

36. જો a અને b ધન પૂર્ણાંક હોય, તો $open parentheses table row bold a row bold b end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 1 end cell end table close parentheses bold space open parentheses table row bold b row bold 1 end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 2 end cell end table close parentheses bold space open parentheses table row bold b row bold 2 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold b row bold r end table close parentheses bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold. bold space$

$open parentheses table row cell bold a bold plus bold b end cell row bold r end table close parentheses$
$open parentheses table row cell bold a bold plus bold b end cell row cell bold r bold plus bold 1 end cell end table close parentheses$
$fraction numerator bold left parenthesis bold a bold plus bold b bold right parenthesis bold factorial over denominator bold r bold factorial end fraction$
$fraction numerator bold a bold factorial bold b bold factorial over denominator bold r bold factorial end fraction$

open parentheses table row cell bold a bold plus bold b end cell row bold r end table close parentheses

Tips: -

(1 + x)^a (1 + x)^b = (1 + x)^a+b નું બંને બાજુ વિસ્તરણ કરતાં,

$open square brackets open parentheses table row bold a row bold 0 end table close parentheses bold plus open parentheses table row bold a row bold 1 end table close parentheses bold space bold x bold space bold plus bold space open parentheses table row bold a row bold 2 end table close parentheses bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold a row bold a end table close parentheses bold space bold x to the power of bold a close square brackets bold space open square brackets open parentheses table row bold b row bold 0 end table close parentheses bold plus open parentheses table row bold b row bold 1 end table close parentheses bold space bold x bold space bold plus bold space open parentheses table row bold b row bold 2 end table close parentheses bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold b row bold b end table close parentheses bold space bold x to the power of bold b close square brackets bold equals bold space open parentheses table row cell bold a bold plus bold b end cell row bold 0 end table close parentheses bold space bold plus bold space open parentheses table row cell bold a bold plus bold b end cell row bold 1 end table close parentheses bold space bold x bold space bold plus bold space open parentheses table row cell bold a bold plus bold b end cell row bold 2 end table close parentheses bold space bold x to the power of bold 2 bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row cell bold a bold plus bold b end cell row cell bold a bold plus bold b end cell end table close parentheses bold space bold x to the power of bold a bold plus bold b end exponent$

બંને બાજુ x^r નો સહગુણક સરખાવતાં,

$bold therefore bold space open parentheses table row bold a row bold r end table close parentheses bold space open parentheses table row bold b row bold 0 end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 1 end cell end table close parentheses bold space open parentheses table row bold b row bold 1 end table close parentheses bold space bold plus bold space open parentheses table row bold a row cell bold r bold minus bold 2 end cell end table close parentheses bold space open parentheses table row bold b row bold 2 end table close parentheses bold space bold plus bold space bold. bold. bold. bold space bold plus bold space open parentheses table row bold a row bold 0 end table close parentheses bold space open parentheses table row bold b row bold r end table close parentheses bold space bold equals bold space open parentheses table row cell bold a bold plus bold b end cell row bold r end table close parentheses$

$bold therefore$ માંગેલ સરવાળો $bold equals bold space open parentheses table row cell bold a bold plus bold b end cell row bold r end table close parentheses$