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Using the digits 2, 3, 4, 7 and 8, Carlos will form five-digit positive integers. Only the digit 2 can be used more than once in any of Carlos’ five-digit integers. How many distinct fivedigit positive integers are possible?

 Jan 11, 2021
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Using 2 once  =   all arrangements of 5 different digits  =    5!  =   120

 

Using 2  twice  =   choose  any two  positions for the 2s,  = C(5,2).....choose any 3 of the remaining 4 digits =C(4,3)...and arrange them in 3! ways =  C(5,2) * C(4,3) * 3!   =   240

 

Using 2 three times =  choose any 3 poisitions for  the 2s  = C(5, 3) ....choose  any 2 of the  4  remaining digits  = C(4,2)  and arrange these in 2!  ways =  C(5,3) * C(4,2) * 2!   =  120

 

Using 2 four times  =   choose any  4 of 5 positions for the 2s =  C(5,4)   and  for the  other position,  choose  any  1 of the remaining  4 digits...C(5,4) * C(4,1)  = 20

 

Using all 5  2s =   1 

 

120  + 240  +  120  + 20  +  1  = 

 

501   numbers

 

 

cool cool cool 

 Jan 11, 2021

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