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Gujarati JEE Mathematics : ક્રમચય અને સંચય

Multiple Choice Questions

1. 100 વ્યક્તિઓમાંથી 20 ચોક્કસ વ્યક્તિઓ આવે જ તે રીતે 30 વ્યક્તિઓને એક હારમાં કેટલી રીતે ગોઠવી શકાય ?

$open parentheses table row bold 100 row bold 20 end table close parentheses open parentheses table row bold 80 row bold 10 end table close parentheses bold space bold 30 bold factorial$
$open parentheses table row bold 80 row bold 10 end table close parentheses bold space bold 30 bold factorial$
20 × ₈₀ P ₁₀
₈₀ P ₁₀ × 30!

2. જો $bold m bold space bold equals bold space bold sum from bold r bold equals bold 0 to bold 100 of bold space fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold r end table close parentheses end fraction bold comma$ તો $bold sum from bold r bold equals bold 0 to bold 100 of bold space fraction numerator bold r over denominator open parentheses table row bold 100 row bold r end table close parentheses end fraction bold space bold equals bold space bold. bold. bold. bold. bold. bold. bold space$

100 m
200 m
50 m
99 m

50 m

Tips: -

ધારો કે $bold I bold space bold equals bold space bold sum from bold r bold equals bold 0 to bold 100 of bold space fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold r end table close parentheses end fraction$

$bold equals bold space fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 0 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold 1 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 2 over denominator open parentheses table row bold 100 row bold 2 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 3 over denominator open parentheses table row bold 100 row bold 3 end table close parentheses end fraction bold space bold plus bold space bold. bold. bold. bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 100 end table close parentheses end fraction bold space bold space bold space bold space bold space bold space bold space bold. bold. bold space bold left parenthesis bold 1 bold right parenthesis bold space bold equals bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 100 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 99 over denominator open parentheses table row bold 100 row bold 99 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 98 over denominator open parentheses table row bold 100 row bold 98 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 97 over denominator open parentheses table row bold 100 row bold 97 end table close parentheses end fraction bold space bold plus bold space bold. bold. bold. bold plus fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 0 end table close parentheses end fraction$

$bold therefore bold space bold I bold space bold equals bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 0 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 99 over denominator open parentheses table row bold 100 row bold 1 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 98 over denominator open parentheses table row bold 100 row bold 2 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 97 over denominator open parentheses table row bold 100 row bold 3 end table close parentheses end fraction bold plus bold. bold. bold. bold space bold plus bold space fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 100 end table close parentheses end fraction bold space bold space bold space open parentheses table row bold n row bold r end table close parentheses bold space bold equals bold space open parentheses table row bold n row cell bold n bold minus bold r end cell end table close parentheses bold space bold. bold. bold. bold. bold. bold. bold space bold left parenthesis bold 2 bold right parenthesis bold space bold left parenthesis bold 1 bold right parenthesis bold space bold plus bold space bold left parenthesis bold 2 bold right parenthesis bold space bold rightwards double arrow bold 2 bold I bold space bold equals bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 0 end table close parentheses end fraction bold space bold plus bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 1 end table close parentheses end fraction bold plus bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 2 end table close parentheses end fraction bold plus bold space fraction numerator bold 100 over denominator open parentheses table row bold 100 row bold 3 end table close parentheses end fraction bold plus bold. bold. bold. bold plus fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 100 end table close parentheses end fraction bold space bold space bold space bold equals bold 100 bold space open square brackets fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold 0 end table close parentheses end fraction bold plus fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold 1 end table close parentheses end fraction bold plus fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 2 end table close parentheses end fraction bold plus fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold 3 end table close parentheses end fraction bold plus bold. bold. bold. bold plus fraction numerator bold 1 over denominator open parentheses table row bold 100 row bold 100 end table close parentheses end fraction close square brackets bold space bold space bold space bold equals bold space bold 100 bold space bold sum from bold r bold equals bold 0 to bold 100 of bold space bold space fraction numerator bold 0 over denominator open parentheses table row bold 100 row bold r end table close parentheses end fraction bold space bold space bold space bold space bold equals bold space bold 100 bold space bold m bold space bold therefore bold space bold I bold space bold equals bold space bold 50 bold space bold m bold space$

3. 50 વ્યક્તિઓમાંથી ચોક્કસ વ્યક્તિઓ આવે જ, તે રીતે 30 વ્યક્તિઓની પસંદગી કુલ કેટલી રીતે કરી શકાય ?

$open parentheses table row bold 46 row bold 26 end table close parentheses$
$open parentheses table row bold 46 row bold 30 end table close parentheses$
₄₆ P ₂₆
$open parentheses table row bold 46 row bold 26 end table close parentheses$

4. m + n + p વસ્તુઓમાંથી m, n અને p વસ્તુઓને ત્રણ ખાનામાં કુલ ........ રીતે મૂકી શકાય ?

$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold space bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p bold space bold factorial end fraction bold space bold cross times bold space bold 3$
$fraction numerator bold left parenthesis bold m bold space bold space bold n bold space bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p bold space bold factorial end fraction bold space bold cross times bold space bold 1 over bold 6$
$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold space bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p bold space bold factorial end fraction$
$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold space bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p bold space bold factorial end fraction bold space bold cross times bold space bold 3 bold space bold factorial$

n વ્યક્તિઓમાંથી 5 ચોક્કસ તક્તિઓ ન આવે, તે રીતે 80 વ્યક્તિઓને વર્તુળાકાર ટેબલ પર કેટલી રીતે ગોઠવી શકાય ?

$open parentheses table row cell bold n bold minus bold 5 end cell row cell bold n bold minus bold 85 end cell end table close parentheses bold space bold 79 bold space bold factorial$
$1 half space open parentheses table row cell bold n bold minus bold 5 end cell row bold 80 end table close parentheses bold space bold 79 bold space bold factorial$
$bold P presubscript bold left parenthesis bold n bold minus bold 5 bold right parenthesis end presubscript subscript bold 80$
$open parentheses table row cell bold n bold minus bold 5 end cell row bold 80 end table close parentheses bold space bold 80 bold space bold factorial$

6. MONDAY શબ્દના અક્ષરોની મદદથી Y ન આવે તે રીતે ચાર અક્ષરના કુલ કેટલા શબ્દો ન બને ?

7. કુલ m + n + p વસ્તુઓનું m, વસ્તુઓ, n વસ્તુઓ અને p વસ્તુઓના ત્રણ ભાગમાં વિભાજન કેટલી રીતે કરી શકાય ?

$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p end fraction bold cross times bold space bold 3$
$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p end fraction bold cross times bold space bold 1 over bold 6$
$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p end fraction bold cross times bold space bold 3 bold space bold factorial$
$fraction numerator bold left parenthesis bold m bold space bold plus bold space bold n bold plus bold space bold p bold right parenthesis bold space bold factorial over denominator bold m bold space bold factorial bold space bold n bold space bold factorial bold space bold p bold space bold factorial end fraction$

8. mn ભિન્ન વસ્તુઓનું n સમાન ભાગમાં ........ રીતે વિભાજન કરી શકાય.

$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction bold cross times bold space fraction numerator bold 1 over denominator bold n bold space bold factorial end fraction$
$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction$
$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction bold cross times bold 1 over bold n$
આપેલ પૈકી એક પણ નહી

9. mn ભિન્ન વસ્તુઓ n વ્યક્તિઓને સમાન ભાગે ....... રીતે વહેંચી શકાય.

$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction$
$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction bold cross times bold space bold n bold space bold factorial$
$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold n bold space bold factorial bold space bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction$
$fraction numerator bold left parenthesis bold m bold space bold n bold right parenthesis bold space bold factorial over denominator bold left parenthesis bold m bold space bold factorial bold right parenthesis to the power of bold n end fraction bold space bold cross times bold space bold n bold space bold factorial$