Permutations And Combinations

Question
CBSEENMA11014104

How many 4-digit numbers are there if no digit is repeated?

Or

How many numbers are there between 1000 and 9999 so that no digit is repeated?

Solution

Number of digits available = 10
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
Number of digits to be used = 4

Since the numbers are to be of four digits.
∴    0 cannot be filled in box 1.
      Number of permutations for box 1 = 9                                                         ...(i)
      Now, we have 9-digits and 3 boxes.
rightwards double arrow                       n = 9,  r = 3.
rightwards double arrow   Number of permutations = straight P presuperscript 9 subscript 3 space equals space fraction numerator 9 factorial over denominator 6 factorial end fraction space equals space fraction numerator 9 cross times 8 cross times 7 cross times 6 factorial over denominator 6 factorial end fraction equals 9 cross times 8 cross times 7 space equals space 504   ...(ii)
       From (i) and (ii), we have total number of permutations, by fundamentals principle of counting
                                 = 9 x 504 = 4536.

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