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Question

# How many 5 digit even numbers can be made from the digits 0,1,2,3,4,5 if repetition is not allowed?

A

96

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B

120

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C

312

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D

360

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Solution

## The correct option is C 312 Even numbers can be formed by placing either 0 or 2 or 4 in the rightmost digit. _ _ _ _ 0 _ _ _ _ 2 _ _ _ _ 4 So, let us consider all the 3 cases separately. Case 1: Rightmost digit is 0 Available digits - 1,2,3,4,5 Leftmost digit can be filed with 1 or 2 or 3 or 4 or 5 So, number if ways of filling leftmost digit = 5 As 2 digits are already used, the next position can be filled with either of the 4 digits. Proceeding in this manner, the next 2 digits can be filled in 3 and 2 ways respectively −|5−|4−|3_0|2 So, the number of possible numbers = 5×4×3×2 = 120 Case 2: Rightmost digit is 2 _ _ _ _ 2 Available digits 0,1,3,4,5 Out of these 5 digits leftmost digit can be filed with 1 or 3 or 4 or 5 As if 0 comes on the left most position the number will become a 4 digit number. So, number if ways of filling leftmost digit = 4 As 2 digits are already used, the next position can be filled with either of the 4 digits . Proceeding in this manner, the next 2 digits can be filled in 3 and 2 ways respectively. −|4−|4−|3_2|2 So, the number of possible numbers = 4×4×3×2 = 96 Case 3: Rightmost digit is 4 This case is similar to 2nd case. −|4−|4−|3_4|2 Number of possible numbers = 4×4×3×2 = 96 ∴ the total number of arrangements = 120 + 96 + 96 = 312

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