How many positive integers are there with distinct digits??
How many positive integers are there with distinct digits??
First, every one-digit number has all of its digits distinct. (This is obvious.) So that’s 9.
For two-digit numbers, we have nine options for the first digit and nine options for the second (the first digit can’t be 0 or we just have a one-digit number; the second digit can’t be the same as the first)no.of 2 digit nos.=9x9.= 81
The process continues similarly - three-digit numbers have 9x9x8=648 possibilities;
four-digit numbers have 9x9x8x7=4536;
five-digit numbers have 9x9x8x7x6=27216;
six-digit numbers have 9x9x8x7x6x5=136080;
seven-digit numbers have 9x9x8x7x6x5x4=544320;
eight-digit numbers have 9x9x8x7x6x5x4x3=1632960;
nine-digit numbers have 9x9x8x7x6x5x4x3x2=3265920;
and ten-digit numbers have 9x9x8x7x6x5x4x3x2x1=3265920 possibilities. (Since we’re working in base 10, any number with 11 or more digits has at least one repeated digit.)
The grand total, therefore, is 9+81+648+4536+27216+136080+544320+1632960+3265920+3265920 = 8877690
First, every one-digit number has all of its digits distinct. (This is obvious.) So that’s 9.
For two-digit numbers, we have nine options for the first digit and nine options for the second (the first digit can’t be 0 or we just have a one-digit number; the second digit can’t be the same as the first)no.of 2 digit nos.=9x9.= 81
The process continues similarly - three-digit numbers have 9x9x8=648 possibilities;
four-digit numbers have 9x9x8x7=4536;
five-digit numbers have 9x9x8x7x6=27216;
six-digit numbers have 9x9x8x7x6x5=136080;
seven-digit numbers have 9x9x8x7x6x5x4=544320;
eight-digit numbers have 9x9x8x7x6x5x4x3=1632960;
nine-digit numbers have 9x9x8x7x6x5x4x3x2=3265920;
and ten-digit numbers have 9x9x8x7x6x5x4x3x2x1=3265920 possibilities. (Since we’re working in base 10, any number with 11 or more digits has at least one repeated digit.)
The grand total, therefore, is 9+81+648+4536+27216+136080+544320+1632960+3265920+3265920 = 8877690
