Question and Answers Forum

All Questions      Topic List

Permutation and Combination Questions

Previous in All Question      Next in All Question      

Previous in Permutation and Combination      Next in Permutation and Combination      

Question Number 124400 by mr W last updated on 03/Dec/20

How many six-digit numbers contain  exactly three different digits?

$${How}\:{many}\:{six}-{digit}\:{numbers}\:{contain} \\ $$$${exactly}\:{three}\:{different}\:{digits}? \\ $$

Answered by benjo_mathlover last updated on 03/Dec/20

XXXWYZ = 9×C_( 3) ^( 9)  × ((6!)/(3!))  =9×((9×8×7)/(3×2×1)) ×6×5×4=90720   where X≠ 0

$${XXXWYZ}\:=\:\mathrm{9}×{C}_{\:\mathrm{3}} ^{\:\mathrm{9}} \:×\:\frac{\mathrm{6}!}{\mathrm{3}!} \\ $$$$=\mathrm{9}×\frac{\mathrm{9}×\mathrm{8}×\mathrm{7}}{\mathrm{3}×\mathrm{2}×\mathrm{1}}\:×\mathrm{6}×\mathrm{5}×\mathrm{4}=\mathrm{90720} \\ $$$$\:{where}\:{X}\neq\:\mathrm{0} \\ $$$$ \\ $$

Commented by benjo_mathlover last updated on 03/Dec/20

what wrong sir?   in part XXXYYZ ?

$${what}\:{wrong}\:{sir}?\: \\ $$$${in}\:{part}\:{XXXYYZ}\:? \\ $$

Answered by mr W last updated on 03/Dec/20

to select three digits there are   C_3 ^(10) =120 ways.  say the three digits are x,y,z.  to form a 6 digit number with these  three digits we have  case 1: 4x+y+z  ⇒3×((6!)/(4!))=90  case 2: 3x+2y+z  ⇒3×2×((6!)/(3!2!))=360  case 3: 2x+2y+2z  ⇒((6!)/(2!2!2!))=90  ⇒120×(90+360+90)=64800    but in these numbers some begin  with zero:  0xxxxy ⇒2×((5!)/(4!))=10  0xxxyy ⇒2×((5!)/(3!2!))=20  0xxxy0 ⇒2×((5!)/(3!))=40  0xxyy0 ⇒((5!)/(2!2!))=30  0xxy00 ⇒2×((5!)/(2!2!))=60  0xy000 ⇒((5!)/(3!))=20  to select the two digits x and y  there are C_2 ^9 =36 ways.  ⇒36×(10+20+40+30+60+20)=6480    total valid 6 digit numbers:  64800−6480=58320

$${to}\:{select}\:{three}\:{digits}\:{there}\:{are}\: \\ $$$${C}_{\mathrm{3}} ^{\mathrm{10}} =\mathrm{120}\:{ways}. \\ $$$${say}\:{the}\:{three}\:{digits}\:{are}\:{x},{y},{z}. \\ $$$${to}\:{form}\:{a}\:\mathrm{6}\:{digit}\:{number}\:{with}\:{these} \\ $$$${three}\:{digits}\:{we}\:{have} \\ $$$${case}\:\mathrm{1}:\:\mathrm{4}{x}+{y}+{z} \\ $$$$\Rightarrow\mathrm{3}×\frac{\mathrm{6}!}{\mathrm{4}!}=\mathrm{90} \\ $$$${case}\:\mathrm{2}:\:\mathrm{3}{x}+\mathrm{2}{y}+{z} \\ $$$$\Rightarrow\mathrm{3}×\mathrm{2}×\frac{\mathrm{6}!}{\mathrm{3}!\mathrm{2}!}=\mathrm{360} \\ $$$${case}\:\mathrm{3}:\:\mathrm{2}{x}+\mathrm{2}{y}+\mathrm{2}{z} \\ $$$$\Rightarrow\frac{\mathrm{6}!}{\mathrm{2}!\mathrm{2}!\mathrm{2}!}=\mathrm{90} \\ $$$$\Rightarrow\mathrm{120}×\left(\mathrm{90}+\mathrm{360}+\mathrm{90}\right)=\mathrm{64800} \\ $$$$ \\ $$$${but}\:{in}\:{these}\:{numbers}\:{some}\:{begin} \\ $$$${with}\:{zero}: \\ $$$$\mathrm{0}{xxxxy}\:\Rightarrow\mathrm{2}×\frac{\mathrm{5}!}{\mathrm{4}!}=\mathrm{10} \\ $$$$\mathrm{0}{xxxyy}\:\Rightarrow\mathrm{2}×\frac{\mathrm{5}!}{\mathrm{3}!\mathrm{2}!}=\mathrm{20} \\ $$$$\mathrm{0}{xxxy}\mathrm{0}\:\Rightarrow\mathrm{2}×\frac{\mathrm{5}!}{\mathrm{3}!}=\mathrm{40} \\ $$$$\mathrm{0}{xxyy}\mathrm{0}\:\Rightarrow\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{30} \\ $$$$\mathrm{0}{xxy}\mathrm{00}\:\Rightarrow\mathrm{2}×\frac{\mathrm{5}!}{\mathrm{2}!\mathrm{2}!}=\mathrm{60} \\ $$$$\mathrm{0}{xy}\mathrm{000}\:\Rightarrow\frac{\mathrm{5}!}{\mathrm{3}!}=\mathrm{20} \\ $$$${to}\:{select}\:{the}\:{two}\:{digits}\:{x}\:{and}\:{y} \\ $$$${there}\:{are}\:{C}_{\mathrm{2}} ^{\mathrm{9}} =\mathrm{36}\:{ways}. \\ $$$$\Rightarrow\mathrm{36}×\left(\mathrm{10}+\mathrm{20}+\mathrm{40}+\mathrm{30}+\mathrm{60}+\mathrm{20}\right)=\mathrm{6480} \\ $$$$ \\ $$$${total}\:{valid}\:\mathrm{6}\:{digit}\:{numbers}: \\ $$$$\mathrm{64800}−\mathrm{6480}=\mathrm{58320} \\ $$

Commented by mr W last updated on 04/Dec/20

there is also a formula:  (9/(10))×C_m ^(10) ×m!×{_m ^n }  with n=6 and m=3:  (9/(10))×C_3 ^(10) ×3!×{_3 ^6 }=(9/(10))×120×6×90  =58320

$${there}\:{is}\:{also}\:{a}\:{formula}: \\ $$$$\frac{\mathrm{9}}{\mathrm{10}}×{C}_{{m}} ^{\mathrm{10}} ×{m}!×\left\{_{{m}} ^{{n}} \right\} \\ $$$${with}\:{n}=\mathrm{6}\:{and}\:{m}=\mathrm{3}: \\ $$$$\frac{\mathrm{9}}{\mathrm{10}}×{C}_{\mathrm{3}} ^{\mathrm{10}} ×\mathrm{3}!×\left\{_{\mathrm{3}} ^{\mathrm{6}} \right\}=\frac{\mathrm{9}}{\mathrm{10}}×\mathrm{120}×\mathrm{6}×\mathrm{90} \\ $$$$=\mathrm{58320} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com