Question
The total number of 5-digit numbers of different digits in which the digit in the middle is the largest is
A.
$$\sum\limits_{n = 4}^9 {^n{P_4}} $$
B.
$$33\left( {3!} \right)$$
C.
$$30\left( {3!} \right)$$
D.
None of these
Answer :
None of these
Solution :
The number of numbers with 4 in the middle $${ = ^4}{P_4}{ - ^3}\,{P_3}$$
( $$\because $$ the other four places are to be filled by 0, 1, 2 and 3, and a number cannot begin with 0).
Similarly, the number of numbers with 5 in the middle $$ = {\,^5}{P_4} - {\,^4}{P_3},\,{\text{e}}{\text{.t}}{\text{.c}}{\text{.}}$$
∴ the required number of numbers
$$ = \,\left( {^4{P_4} - {\,^3}{P_3}} \right) + \left( {^5{P_4} - {\,^4}{P_3}} \right) + \left( {^6{P_4} - {\,^5}{P_3}} \right) + ..... + \left( {^9{P_4} - {\,^8}{P_3}} \right).$$