Question:
The number of ways in which a cricket team of 11 players can be chosen out of a batch of 15 players so that the captain of the team is always included, is:
15 खिलाड़ियों के समूह में से 11 खिलाड़ियों की एक क्रिकेट टीम को चुनने के तरीकों की संख्या, ताकि टीम का कप्तान हमेशा शामिल रहे, है:
Options:
- 165
- 364
- 1001
- 1365
- 165
- 364
- 1001
- 1365
Correct Answer: c)
Explanation:
Since the captain must always be included, 1 player is already fixed.
We need to choose the remaining \(11 – 1 = 10\) players from the remaining \(15 – 1 = 14\) players.
Number of ways:
\[{}^{14}C_{10} = {}^{14}C_4 = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} = 14 \times 13 \times 11 / 2 = 7 \times 143 = 1001\]
Hence, the correct option is (c).
चूँकि कप्तान हमेशा शामिल रहेगा, अतः 1 खिलाड़ी पहले से तय है।
शेष \(15 – 1 = 14\) खिलाड़ियों में से शेष \(11 – 1 = 10\) खिलाड़ी चुनने होंगे:
\[{}^{14}C_{10} = {}^{14}C_4 = \frac{14 \times 13 \times 12 \times 11}{24} = 1001\]
अतः सही विकल्प (c) है।