Question:
The number of ways in which any four letters can be selected from the word ‘CORGOO’ is:
‘CORGOO’ शब्द से किन्हीं चार अक्षरों को चुनने के तरीकों की संख्या है:
Options:
- 15
- 11
- 7
- None of these
- 15
- 11
- 7
- इनमें से कोई नहीं
Correct Answer: c)
Explanation:
The word ‘CORGOO’ consists of letters: C(1), R(1), G(1), and O(3).
The distinct letters available are {C, R, G, O}.
We need to select 4 letters. We consider three mutually exclusive cases:
1. All 4 letters are distinct: Select 4 from {C, R, G, O} \(\implies {}^4C_4 = 1\) way.
2. 2 alike (OO) and 2 different: Select ‘OO’ (1 way) and 2 letters from {C, R, G} \(\implies {}^3C_2 = 3\) ways.
3. 3 alike (OOO) and 1 different: Select ‘OOO’ (1 way) and 1 letter from {C, R, G} \(\implies {}^3C_1 = 3\) ways.
Total number of selections \(= 1 + 3 + 3 = 7\).
Hence, the correct option is (c).
‘CORGOO’ में अक्षर: C(1), R(1), G(1), O(3) हैं।
चार अक्षर चुनने की स्थितियाँ:
1. चारों अक्षर भिन्न: {C, R, G, O} से 4 अक्षर चुनना = \({}^4C_4 = 1\)
2. 2 समान (OO) और 2 भिन्न: {C, R, G} से 2 अक्षर = \({}^3C_2 = 3\)
3. 3 समान (OOO) और 1 भिन्न: {C, R, G} से 1 अक्षर = \({}^3C_1 = 3\)
कुल तरीके = \(1 + 3 + 3 = 7\)।
अतः सही विकल्प (c) है।