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Q2:
How many words, with or without meaning, can be formed using all the letters of the word EQUATION at a time so that the vowels and consonants occur together?
Solution :
Step 1: Decomposition of the Given Word
The word EQUATION consists of $8$ distinct letters:
- Vowels ($V$): {E, U, A, I, O} — Total $n_v = 5$
- Consonants ($C$): {Q, T, N} — Total $n_c = 3$
Step 2: Grouping Strategy
The problem requires vowels to occur together and consonants to occur together. We treat the set of vowels as one single unit ($V_{unit}$) and the set of consonants as one single unit ($C_{unit}$).
Total items to arrange = $\{V_{unit}, C_{unit}\}$.
Number of ways to arrange these 2 units = $2!$ [By the Permutation Theorem $n!$ for $n$ distinct items].
Step 3: Internal Permutations
A. Arranging Vowels internally: The 5 distinct vowels {E, U, A, I, O} can be arranged among themselves in $5!$ ways.
Calculation: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ ways.
B. Arranging Consonants internally: The 3 distinct consonants {Q, T, N} can be arranged among themselves in $3!$ ways.
Calculation: $3! = 3 \times 2 \times 1 = 6$ ways.
Step 4: Total Calculation
Applying the Multiplication Principle [If an event can occur in $m$ ways and another independent event in $n$ ways, the total number of occurrences is $m \times n$]:
Total Words = (Ways to arrange units) $\times$ (Internal arrangement of Vowels) $\times$ (Internal arrangement of Consonants)
Total Words = $2! \times 5! \times 3!$
Total Words = $2 \times 120 \times 6$
Total Words = $1440$
Final Solution: The total number of words that can be formed is $1440$.
More Questions from Class 11 Mathematics Permutations and Combinations Miscellaneous Exercise on Chapter 6
- Q1: How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER ?
- Q3(i): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls ?
- Q3(ii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (ii) atleast 3 girls ?
- Q3(iii): A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (iii) atmost 3 girls ?
- Q4: If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E ?
- Q5: How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated ?
- Q6: The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet ?
- Q7: In an examination, a question paper consists of 12 questions divided into two parts i.e., Part I and Part II, containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?
- Q8: Determine the number of 5-card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.
- Q9: It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How many such arrangements are possible ?
- Q10: From a class of 25 students, 10 are to be chosen for an excursion party. There are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen ?
- Q11: In how many ways can the letters of the word ASSASSINATION be arranged so that all the $S$'s are together ?
CBSE Solutions for Class 11 Mathematics Permutations and Combinations
Chapters in CBSE - Class 11 Mathematics
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