Q. How many 8-digit even numbers less than 3,000,000 can be formed using the following digits: 1, 2, 2, 3, 5, 5, 6, 6? Choose the correct choice from the following answers.
A) 180
B) 240
C) 210
D) 270
E) 300
the number is less than 3000000 so we will have the digit on the left most side to be either 1 or 2   i.e your number can be 1 _ _ _ _ _ _ or 2 _ _ _ _ _ _   case 1 &gt; 1 _ _ _ _ _ _ here the units digit can be either 2 or 6   when the units digit is 2 i.e 1 _ _ _ _ _ 2   number of ways this can be done would be 5! / 2! (as 5 is repeated twice) = 60   when the units digit is 6 i.e. 1 _ _ _ _ _ 6, number of ways would be 5!/(2! * 2!) {both 2 and 5 repeat twice} = 30   case 2 &gt; 2 _ _ _ _ _ _ (here units digit may be 2 or 6)   number of ways this can be done would be 5! / (2!) for 2 =60   and 5! / 2! for 6 ... = 60   adding all of these ... gives the answer 270 ...   the explanation looks difficult but this technique helps if you are familiar with the concept of permutations when the numbers/alphabets repeat   270
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Q. How many 8-digit even numbers less than 3,000,000 can be formed using the following digits: 1, 2, 2, 3, 5, 5, 6, 6?