How many five letter words (with or without meaning) can be formed such that the letters appearing in the odd positions are taken from the unrepeated letters of the word MATHEMATICS whereas the letters which occupy even places are taken from amongst the repeated letters of the same?Choices:- 480 540 460 520

The word MATHEMATICS has 11 letters where M, A, T, I are repeated.

Step 1: Identify the unique and repeated letters. The unique letters are: M, A, T, H, E, I, C, S. So, we have 8 unique letters. The repeated letters are: M, A, T, I. So, we have 4 repeated letters.

Step 2: Determine the number of ways to arrange the letters in odd positions. A 5-letter word has 3 odd positions (1st, 3rd, 5th). We are choosing 3 letters from 8 unique letters. This can be done in 8P3 ways = 876 = 336 ways.

Step 3: Determine the number of ways to arrange the letters in even positions. A 5-letter word has 2 even positions (2nd, 4th). We are choosing 2 letters from 4 repeated letters. This can be done in 4P2 ways = 4*3 = 12 ways.

Step 4: Multiply the number of ways to arrange the letters in odd and even positions. The total number of ways to form the word is 336 * 12 = 4032.

However, this is not one of the options given. It seems there might be a mistake in the question or the options provided.

The word MATHEMATICS has 11 letters where M, A, T, I are repeated.

Step 1: Identify the unique and repeated letters in the word MATHEMATICS.

Unique letters: M, A, T, H, E, I, C, S Repeated letters: M, A, T, I

Step 2: We need to form a 5-letter word. The letters in the odd positions (1st, 3rd, 5th) are taken from the unique letters, and the letters in the even positions (2nd, 4th) are taken from the repeated letters.

Step 3: There are 3 odd positions and 2 even positions in the 5-letter word.

Step 4: The number of ways to fill the odd positions is the number of permutations of 8 unique letters taken 3 at a time, which is 8P3 = 876 = 336.

Step 5: The number of ways to fill the even positions is the number of permutations of 4 repeated letters taken 2 at a time, which is 4P2 = 4*3 = 12.

Step 6: The total number of 5-letter words is the product of the number of ways to fill the odd and even positions, which is 336*12 = 4032.

So, none of the given options (480, 540, 460, 520) is correct. The correct answer is 4032.