The correct answer is Option (4) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)
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List-I
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List-II
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(A) The number of different words that can be formed with CUSTOM with the condition that the word should begin with M is _____.
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(IV) 120
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(B) The number of different ways in which the letters of the word EXTRA can be arranged so that the vowels are never together is _____.
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(III) 72
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(C) There are 10 points in a plane. No three of these points are in a straight line. The total number of straight line that can be formed by joining the two points is ______.
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(II) 45
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(D) The number of ways a committee of 4 people be chosen out of 8 people is _____.
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(I) 70
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(A) Words from "CUSTOM" beginning with 'M'
The word CUSTOM has 6 distinct letters: C, U, S, T, O, M.
- If the word must begin with M, we fix 'M' in the first position: M _ _ _ _ _
- The remaining 5 positions can be filled by the remaining 5 letters (C, U, S, T, O) in $5!$ ways.
- Calculation: $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
- Match: (A) $\rightarrow$ (IV)
(B) "EXTRA" arrangements where vowels are never together
The word EXTRA has 5 distinct letters: E, X, T, R, A.
- Total arrangements: $5! = 120$.
- Arrangements with vowels together: Treat the vowels (E, A) as a single unit. We now have 4 units: (EA), X, T, R.
- Ways to arrange these 4 units: $4! = 24$.
- Internal arrangements of the vowels within the unit: $2! = 2$.
- Total "together" cases: $24 \times 2 = 48$.
- Vowels never together: $\text{Total} - \text{Together} = 120 - 48 = 72$.
- Match: (B) $\rightarrow$ (III)
(C) Straight lines from 10 points
A straight line is formed by joining any 2 points. If no three points are in a straight line (non-collinear), the number of lines is the number of ways to choose 2 points out of 10.
- Formula: ${}^{10}C_2 = \frac{10 \times 9}{2 \times 1} = 45$.
- Match: (C) $\rightarrow$ (II)
(D) Committee of 4 out of 8 people
The number of ways to choose 4 people out of 8 is a combination problem.
- Formula: ${}^8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70$.
- Match: (D) $\rightarrow$ (I)
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