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By using the non-zero digits, the number of 5 digit numbers that can be formed so that each number has largest digit in its middle place and the digits in the number are different is
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Verified Answer
The correct answer is:
$\sum_{r=4}^8{ }_r^r P_4$
Since, the largest digit is in the middle, the middle digit is greater than or equal to 5 , the number of numbers with 5 in the middle $={ }^4 P_4$ ( $\because$ the four places are to be filled by $1,2,3,4$ ) Similarly, the number of numbers with 6 in the middle $={ }^5 P_4$
The number of numbers with 7 in the middle $={ }^6 P_4$
The number of numbers with 8 in the middle $={ }^7 P_4$
The number of numbers with 9 in the middle $={ }^8 P_4$
$\therefore$ Total number of such numbers
$$
={ }^4 P_4+{ }^5 P_4+{ }^6 P_4+{ }^7 P_4+{ }^8 P_4=\sum_{r=4}^8{ }^r P_4
$$
The number of numbers with 7 in the middle $={ }^6 P_4$
The number of numbers with 8 in the middle $={ }^7 P_4$
The number of numbers with 9 in the middle $={ }^8 P_4$
$\therefore$ Total number of such numbers
$$
={ }^4 P_4+{ }^5 P_4+{ }^6 P_4+{ }^7 P_4+{ }^8 P_4=\sum_{r=4}^8{ }^r P_4
$$
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