How many distinct permutations can be made from the letters of the word MATHEMATICS?
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Answer: 1. MATHEMATICS = 11 letters P(n,r) = P(11,11) = 11!/(11−11)! = 3.99168E+7 = 39,916,800 Repeated letters: 2 M, 2 A, 2 T 39,916,800/(2!2!2!) = 4,989,600 ways 2. P(n,r) = P(6,6) = 6!/(6−6)! = 720 ways 3. VIRUS = 5 letters P(n,r) = P(5,3) = 5!/(5−3)! = 60 ways 4. P(n,r) = P(8,8) = 8!/(8−8)! = 40,320 ways 5. P(n,r) = P(11,3) = 11!/(11−3)! = 990 ways Answer: 1. 4,989,600 2. 720 3. 20 4. 42320 5. 165 Answer:
Explanation:
11! (2! 2! 2! 1! 1! 1! 1! 1! ) =1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 {(1 x 2) (1 x 2) (1 x 2) (1) (1) (1) (1) (1)} =39916800 8 = 4989600 Answer Verified
Hint:Here, we will proceed by observing all the letters in the word MATHEMATICS that are repeating and then, we will use the formula i.e., Permutation of n items out of which x items, y items and z items of different types are repeating = $\dfrac{{n!}}{{x!y!z!}}$. For the next two parts, we will fix the first letter of the word as C and T in order to find out the different arrangements possible.Complete step-by-step answer: For the words which begin with letter C formed using all the letters of the word MATHEMATICS, the first letter is fixed as C so the next 10 letters need to be selected from the left letters (i.e., 2 M’s, 2 A’s, 2 T’s, 1 H, 1 E, 1 I and 1 S) For the words which
begin with letter T formed using all the letters of the word MATHEMATICS, the first letter is fixed as T so the next 10 letters need to be selected from the left letters (i.e., 2 M’s, 2 A’s, 1 T, 1 H, 1 E, 1 I, 1 C and 1 S) Note- In this particular problem, since we have to rearrange the letters of the word MATHEMATICS that’ s why we are using permutation formulas. If we were asked for selection of some letters out of all the letters we would have used combinations formula. The general formula for arrangement of r items out of n items is given by \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\]. How many four letter permutations can be formed from the letters of the word MATHEMATICS?Therefore, the number of ways in which four letters of the word MATHEMATICS can be arranged is 2454.
How many letters are there in the word MATHEMATICS?Solution : (i) There are 11 letters in the word 'MATHEMATICS' . Out of these letters M occurs twice, A occurs twice, T occurs twice and the rest are all different.
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