જો  from Mathematics દ્વિપદી પ્રમેય
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Gujarati JEE Mathematics : દ્વિપદી પ્રમેય

Multiple Choice Questions

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31. 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

C.

76880

Tips: -

fraction numerator bold c subscript bold r over denominator bold c subscript bold r bold minus bold 1 end fraction bold space bold equals bold space fraction numerator open parentheses table row bold 30 row bold r end table close parentheses over denominator open parentheses table row bold 30 row cell bold r bold minus bold 1 end cell end table close parentheses end fraction bold space bold equals bold space fraction numerator bold 30 bold space bold factorial over denominator bold left parenthesis bold 30 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 cross times bold space fraction numerator bold left parenthesis bold 30 bold space bold minus bold space bold r bold space bold plus bold space bold 1 bold right parenthesis bold space bold factorial bold space bold left parenthesis bold r bold space bold minus bold space bold 1 bold space bold right parenthesis bold space bold factorial over denominator bold 30 bold space bold factorial end fraction bold space bold equals bold space fraction numerator bold 31 bold space bold minus bold space bold r over denominator bold r end fraction

માગેલ સરવાળો = bold sum from bold r bold equals bold 1 to bold 30 of bold space bold r to the power of bold 3 bold space open parentheses fraction numerator bold c subscript bold r over denominator bold c subscript bold r bold minus bold 1 end fraction close parentheses to the power of bold 2

                  bold equals bold space bold sum from bold r bold equals bold 1 to bold 30 of bold space bold r to the power of bold 3 bold space open parentheses fraction numerator bold 31 bold minus bold r over denominator bold r end fraction close parentheses to the power of bold 2 bold equals bold space bold sum from bold r bold equals bold 1 to bold 30 of bold space bold r bold space bold left parenthesis bold 961 bold space bold minus bold space bold 62 bold r bold space bold plus bold space bold r to the power of bold 2 bold right parenthesis bold space bold equals bold space bold sum from bold r bold equals bold 1 to bold 30 of bold space bold left parenthesis bold 961 bold space bold r bold space bold minus bold space bold 62 bold space bold r to the power of bold 2 bold space bold plus bold space bold r to the power of bold 3 bold right parenthesis bold space bold equals bold space open square brackets bold 961 bold space fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold n end fraction bold space bold minus bold 61 fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis bold left parenthesis bold 2 bold n bold plus bold 1 bold right parenthesis over denominator bold 6 end fraction bold plus fraction numerator bold n to the power of bold 2 bold space bold left parenthesis bold n bold plus bold 1 bold right parenthesis to the power of bold 2 over denominator bold 4 end fraction close square brackets subscript bold n bold space bold equals bold space bold 30 end subscript bold equals bold space open square brackets fraction numerator bold n bold space bold left parenthesis bold n bold space bold plus bold space bold 1 bold right parenthesis over denominator bold 2 end fraction open parentheses bold 961 bold minus fraction numerator bold 124 bold n over denominator bold 3 end fraction bold minus bold 62 over bold 3 bold plus fraction numerator bold n to the power of bold 2 bold plus bold n over denominator bold 2 end fraction close parentheses close square brackets subscript bold n bold equals bold 30 end subscript bold equals bold space open square brackets fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 2 end fraction open parentheses fraction numerator bold 5766 bold minus bold 248 bold n bold minus bold 124 bold plus bold 3 bold n to the power of bold 2 bold plus bold 3 bold n over denominator bold 6 end fraction close parentheses close square brackets subscript bold n bold equals bold 30 end subscript bold equals bold space open square brackets fraction numerator bold n bold left parenthesis bold n bold plus bold 1 bold right parenthesis over denominator bold 12 end fraction bold space bold left parenthesis bold 3 bold n to the power of bold 2 bold space bold minus bold space bold 245 bold n bold space bold plus bold space bold 5642 close square brackets subscript bold n bold space bold equals bold space bold 30 end subscript bold equals bold space bold 155 over bold 2 bold space bold left parenthesis bold 8342 bold space bold minus bold space bold 7350 bold right parenthesis bold space bold equals bold space bold 155 over bold 2 bold space bold left parenthesis bold 992 bold right parenthesis bold space bold equals bold space bold 155 bold space bold cross times bold space bold 496 bold space bold equals bold space bold 76880
Answer
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    32. 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
    Answer
    • 33. (1 + x + x2 + x3 + x4)3 ના વિસ્તરણમાં x5 નો સહગુણક ...... છે.
    • 21
    • 24
    • 18
    • 15
    Answer
    • 34.
    સમીકરણ x + y + z = 20, x ≥ 1, y ≥ 2, z ≥ 3, x, y, z ∈ Wના ઉકેલોની સંખ્યા ...... છે.
    • 124
    • 136
    • 210
    • 120
    Answer
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    35. 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
    Answer
    • 36. અસમતા x + y + z ≤ 30,  ના અનૃપ પૂર્ણાંક ઉકેલોની સંખ્યા ....... છે.
    • 5646
    • 5456
    • 5465
    • 5654
    Answer
    • 37.
    15 સમાન વસ્તુઓને ત્રણ પેટીમાં કુલ કેટલી રીતે મૂકી શકાય ? (દરેક પેટીમાં વધુમાં વધુ 15 વતુ આવી શકે.)
    • 136
    • 315
    • 15P3
    • 91
    Answer
    • 38. અસંખ્ય લાલ, પીળા અને વાદળી રંગના દડામાંથી 12 દડા પસંદ કરવાના પ્રકાર ....... છે. 
    • 55
    • 91
    • 105
    • અસંખ્ય
    Answer
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    39. શબ્દ ASSASSINATION શબ્દના અક્ષરોમાંથી 4 અક્ષરો કુલ કેટલી રીતે પસંદ કરી શકાય ?
    • 80
    • 71
    • 72
    • 60
    Answer
    • 40. સમીકરણ x1 + x2 + x3 + x4 = 31, x1, x2, x3, x4 ∈ W ના ઉકેલોની સંખ્યા ........ છે.
    • open parentheses table row bold 34 row bold 3 end table close parentheses
    • open parentheses table row bold 35 row bold 4 end table close parentheses
    • open parentheses table row bold 33 row bold 2 end table close parentheses
    • open parentheses table row bold 34 row bold 4 end table close parentheses
    Answer
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