First slide

Permutations

Question

The total number of 5-digit numbers of different digits in which the digit in the middle is the largest is

Moderate

A

$\sum_{n = 4}^{9} (^{n} P_{4} -^{n - 1} P_{3})$
B

$\sum_{n = 4}^{9} {}^{n}P_{4}$
C

$\sum_{n = 4}^{9}^{n - 1} P_{3}$
D

None of these

Solution

The smallest number, which can occur in the middle is 4. The number of numbers with 4 in the middle $=^{4} P_{4} -^{3} P_{3}$ .
(∴ The other four places are to be filled by 0, 1, 2 and 3, and a number can not begin with 0)
Similarly, the number of numbers with 5 in the middle = $^{5} P_{4} -^{4} P_{3}$ , etc.
∴ The required number of numbers
$\begin{array}{l} = (^{4} P_{4} -^{3} P_{3}) + (^{5} P_{4} -^{4} P_{3}) + (^{6} P_{4} -^{5} P_{3}) + \dots + (^{9} P_{4} -^{8} P_{3}) \\ = \sum_{n = 4}^{9} (^{n} P_{4} -^{n - 1} P_{3}) \end{array}$

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