 MATHEMATICS is an eleven letter word = 11! There are 2Ms and 2As and 2Es Divide the number of repeating letters = \(\frac{11!}{2!2!2!}\) There is an explanation video available below. Report an Error Ask A Question Download App Explanation VideoAnswer Hint: First find the number of ways in which word ‘Mathematics’ can be written, and then we use permutation formula with repetition which is given as under,Number of permutation of $n$objects with$n$, identical objects of type$1,{n_2}$identical objects of type \[2{\text{ }} \ldots \ldots .,{\text{ }}{n_k}\]identical objects of type $k$ is \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\] Complete step by step solution: Word Mathematics has $11$ letters\[\mathop {\text{M}}\limits^{\text{1}} \mathop {\text{A}}\limits^{\text{2}} \mathop {\text{T}}\limits^{\text{3}} \mathop {\text{H}}\limits^{\text{4}} \mathop {\text{E}}\limits^{\text{5}} \mathop {\text{M}}\limits^{\text{6}} \mathop {\text{A}}\limits^{\text{7}} \mathop {\text{T}}\limits^{\text{8}} \mathop {\text{I}}\limits^{\text{9}} \mathop {{\text{ C}}}\limits^{{\text{10}}} \mathop {{\text{ S}}}\limits^{{\text{11}}} \]In which M, A, T are repeated twice.By using the formula \[\dfrac{{n!}}{{{n_1}!\,{n_2}!.......{n_k}!}}\], first, we have to find the number of ways in which the word ‘Mathematics’ can be written is $ P = \dfrac{{11!}}{{2!2!2!}} \\ = \dfrac{{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1 \times 2 \times 1}} \\ = 11 \times 10 \times 9 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\ = 4989600 \\ $In \[4989600\]distinct ways, the letter of the word ‘Mathematics’ can be written.(i) When vowels are taken together:In the word ‘Mathematics’, we treat the vowels A, E, A, I as one letter. Thus, we have MTHMTCS (AEAI).Now, we have to arrange letters, out of which M occurs twice, T occurs twice, and the rest are different.$\therefore $Number of ways of arranging the word ‘Mathematics’ when consonants are occurring together$ {P_1} = \dfrac{{8!}}{{2!2!}} \\ = \dfrac{{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \\ = 10080 \\ $Now, vowels A, E, I, A, has $4$ letters in which A occurs $2$ times and rest are different.$\therefore $Number of arranging the letter \[ {P_2} = \dfrac{{4!}}{{2!}} \\ = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1}} \\ = 12 \\ \]$\therefore $Per a number of words $ = (10080) \times (12)$In which vowel come together $ = 120960$ways(ii) When vowels are not taken together:When vowels are not taken together then the number of ways of arranging the letters of the word ‘Mathematics’ are$ = 4989600 - 120960 \\ = 4868640 \\ $Note: In this type of question, we use the permutation formula for a word in which the letters are repeated. Otherwise, simply solve the question by counting the number of letters of the word it has and in case of the counting of vowels, we will consider the vowels as a single unit. 1. After multiplying, we have a (b + c) =2. By factoring, we have ab + ac = simplify the rational number 6 48 Improper fraction to mixed number Solve the following problems using completing the square.1. The area of a rectangle is 96 square centimeters. If the length is 4 cm more than the widt … The given universal set: U= {1,2,3,4,5,6,7,8,9,10} A= {1,2,3,4,5} B= {1,3,5,7,9} C= {0,5,10} Answer the following A U B= A U C= A U B U C= A n B= A n … Create the representation of radian measure in the unit circle. Pi/4 and Pi/6 (Negative rotation) how po to with solution po sna A man plays with building blocks where he placed 35 blocks on the first level, 31 blocks on the second level, 27 blocks on the third, and so on. Conti … 1 to 7 answer this pls Factor the following, Show your solution1. 4x⁶ - 3x²2. x² - 144 3. 9x² - 66x + 121Please answer this question thank you
Permutation and Combination
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How many ways can the letters of the word MATHEMATICS be arranged
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