How many 4-digit numbers are there with no digit repeated?

Asked by Pragya Singh | 1 year ago |  68

##### Solution :-

The thousands place of the 4-digit number can be filled with any digit from 1 to 9 and 0 is not included. Thus, number of ways in which thousands place is filled
is 9. Also, the hundreds, tens and units place can be filled with any digit from 0 to 9. Since, te digits cannot be repeated and thousands place is already occupied by the digit. The hundreds, tens and units place can be filled by remaining 9 digits. Thus, there are permutations of 9 different digits taken 3 at a time. Number of 3-digit numbers =

$$^5P_4=\dfrac{5!}{(5-4)!}=\dfrac{5!}{1!}$$

= 1×2×3×4×5=120

$$^9P_3=\dfrac{9!}{(9-3)!}=\dfrac{9!}{6!}$$

$$\dfrac{9\times 8\times 7\times 6!}{6!}$$

= 9×8×7=504

By, multiplication principle, number of 4-digit numbers is $$9\times 504=4536$$

Answered by Abhisek | 1 year ago

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