First slide

Permutations

Question

The number of even numbers greater than 100 that can be formed by the digits 0, 1, 2, 3 (no digit being repeated) is

Moderate

A

20
B

30
C

40
D

None of these

Solution

The numbers are of three or four digits.
To find the number of even numbers of three digits—
The unit’s place must be filled by 0 or 2.
∴ The number of even numbers of three digits (having or not having 0 in hundred’s place) $=^{2} P_{1} \times^{3} P_{2}$ .
But the number of even numbers of three digits having 0 in hundred’s place $=^{2} P_{1}$ ( $∵$ unit’s place is naturally filled by 2 and ten’s place by one of 1 and 3).
∴ The number of even numbers of three digits
$=^{2} P_{1} \times^{3} P_{2} -^{2} P_{1}$ (1)
Similarly, the number of even numbers of four digits
$=^{2} P_{1} \times^{3} P_{3} -^{2} P_{2}$ (2)
Adding (1) and (2), we get the number of even numbers greater than 100
$\begin{array}{l} = (^{2} P_{1} \times^{3} P_{2} -^{2} P_{1}) + (^{2} P_{1} \times^{3} P_{3} -^{2} P_{2}) \\ = 2 \times 3 \times 2 - 2 + 2 \times 3 \times 2 \times 1 - 2 \times 1 = 20 \end{array}$

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