How many consonants are in the word mathematics

If the number of collections of two vowels is $A$, and that of four consonants is $B$, then the answer is $A·B$.

First to calculate $A$. There are four vowels: A, A, E, I. If two A's are selected, then there is only one collection, i.e. (A, A). If only one A is selected, then there are two collections: (A, E), (A, I). If no A is selected, then there is one collection: (E, I). Thus $A = 4$.

Next to calculate $B$. There are seven consonants: M, M, T, T, H, C, S.
Case 1: Three consonants are selected from H, C, S. To complete the collection, one more consonant has to be selected from M, M, T, T, so it is either an M or a T. Thus there are$$ C(3, 3) × 2 = 2 $$ collections.
Case 2: Two consonants are selected from H, C, S. To complete the collection, two more consonants have to be selected from M, M, T, T, so it is can be (M, M), (M, T) or (T, T). Thus there are$$ C(3, 2) × 3 = 9 $$ collections.
Case 3: One consonant is selected from H, C, S. To complete the collection, three more consonants have to be selected from M, M, T, T, so it is can be (M, M, T) or (M, T, T). Thus there are$$ C(3, 1) × 2 = 6 $$ collections.
Case 4: No consonant is selected from H, C, S. Then there is only one collection, i.e. (M, M, T, T).
Therefore, $B = 2 + 9 + 6 + 1 = 18$.

The final answer id $4 × 18 = 72$.

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.

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:
The word MATHEMATICS consists of 2 M’s, 2 A’s, 2 T’s, 1 H, 1 E, 1 I, 1 C and 1 S.
Total number of letters in the word MATHEMATICS = 11
As we know that
Total number of different arrangements of n items out of which x items, y items and z items of different types are repeating = $\dfrac{{n!}}{{x!y!z!}}{\text{ }} \to {\text{(1)}}$
Using the formula given by equation (1), we can write
Total number of different arrangements which can be made by using all the 11 letters in the word MATHEMATICS in which letter M, letter A and letter T are repeating twice = $\dfrac{{11!}}{{2!2!2!}} = \dfrac{{11.10.9.8.7.6.5.4.3.2!}}{{2.1.2.1.2!}} = \dfrac{{11.10.9.8.7.6.5.4.3}}{4} = 4989600$
Therefore, a total of 4989600 words can be formed using all the letters of the word MATHEMATICS.

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)
Using the formula given by equation (1), we can write
Total number of different arrangements which can be made by using all the left 10 letters (except letter C) in the word MATHEMATICS in which letter M, letter A and letter T are repeating twice = $\dfrac{{10!}}{{2!2!2!}} = \dfrac{{10.9.8.7.6.5.4.3.2!}}{{2.1.2.1.2!}} = \dfrac{{10.9.8.7.6.5.4.3}}{4} = 453600$
Therefore, a total of 453600 words which begin with C can be formed using all the letters of the word MATHEMATICS.

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)
Also we know that
Total number of different arrangements of n items out of which x items and y items of different types are repeating = $\dfrac{{n!}}{{x!y!}}{\text{ }} \to {\text{(2)}}$
Using the formula given by equation (2), we can write
Total number of different arrangements which can be made by using all the left 10 letters (except one of the two letters T) in the word MATHEMATICS in which letter M, letter A are repeating twice = $\dfrac{{10!}}{{2!2!}} = \dfrac{{10.9.8.7.6.5.4.3.2!}}{{2.1.2!}} = \dfrac{{10.9.8.7.6.5.4.3}}{2} = 907200$
Therefore, a total of 907200 words which begin with T can be formed using all the letters of the word MATHEMATICS.

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 \[{}^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\].

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