Respuesta :
Answer:
5040
Step-by-step explanation:
To Find : How many different 4-digit personal identification numbers are possible if no digit can be used twice?
Solution :
Since we are given that no digit can be used twice
SO, when one digit is used so next digit will be chosen from the remaining digit an so on .
the digits can be 0,1,2,3,4,5,6,7,8,9
Since these are 10 in number
out of these 10 we need to choose 4 but no digit can be used twice
⇒[tex]_{10}\textrm{C}_1 *_{9}\textrm{C}_1 *_{8}\textrm{C}_1*_{7}\textrm{C}_1 [/tex]
Formula : [tex]_{n}\textrm{C}_r=\frac{n!}{r!(n-r)!}[/tex]
thus using this formula
⇒[tex]\frac{10!}{1!(10-1)!} *\frac{9!}{1!(9-1)!}*\frac{8!}{1!(8-1)!}*\frac{7!}{1!(7-1)!}[/tex]
⇒[tex]\frac{10!}{9!} *\frac{9!}{8!}*\frac{8!}{7!}*\frac{7!}{6!}[/tex]
⇒[tex]10*9*8*7[/tex]
⇒[tex]5040[/tex]
Thus different 5040 number of 4-digit personal identification numbers are possible if no digit can be used twice
