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16 Sep 2014, 00:48
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During the halftime break of a football match, the coach plans to make 3 substitutions. If the team has 11 players, including 2 forwards, what is the probability that neither of the forwards will be substituted?
A. \(\frac{21}{55}\)
B. \(\frac{18}{44}\)
C. \(\frac{28}{55}\)
D. \(\frac{28}{44}\)
E. \(\frac{36}{55}\)
A. \(\frac{21}{55}\)
B. \(\frac{18}{44}\)
C. \(\frac{28}{55}\)
D. \(\frac{28}{44}\)
E. \(\frac{36}{55}\)
16 Sep 2014, 00:49
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Official Solution:
During the halftime break of a football match, the coach plans to make 3 substitutions. If the team has 11 players, including 2 forwards, what is the probability that neither of the forwards will be substituted?
A. \(\frac{21}{55}\)
B. \(\frac{18}{44}\)
C. \(\frac{28}{55}\)
D. \(\frac{28}{44}\)
E. \(\frac{36}{55}\)
The probability that neither forward will be substituted is \(\frac{C^3_9}{C^3_{11}} = \frac{7 * 8 * 9}{9 * 10 * 11} = \frac{28}{55}\). Here, \(C^3_9\) represents the ways to choose 3 players out of the 9 (excluding the 2 forwards), while \(C^3_{11}\) denotes the ways to choose any 3 players out of the full team of 11.
Answer: C
During the halftime break of a football match, the coach plans to make 3 substitutions. If the team has 11 players, including 2 forwards, what is the probability that neither of the forwards will be substituted?
A. \(\frac{21}{55}\)
B. \(\frac{18}{44}\)
C. \(\frac{28}{55}\)
D. \(\frac{28}{44}\)
E. \(\frac{36}{55}\)
The probability that neither forward will be substituted is \(\frac{C^3_9}{C^3_{11}} = \frac{7 * 8 * 9}{9 * 10 * 11} = \frac{28}{55}\). Here, \(C^3_9\) represents the ways to choose 3 players out of the 9 (excluding the 2 forwards), while \(C^3_{11}\) denotes the ways to choose any 3 players out of the full team of 11.
Answer: C
28 Feb 2016, 08:24
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Bunuel
Probability approach.
First selection. The probability that no one of the forwards be selected from pool of 11 players means that will be selected any from other 9 players = 9/11.
Second selection. The probability that no one of the forwards be selected from pool of 10 remained players means that will be selected any from other remained 8 players = 8/10.
Third selection. The probability that no one of the forwards be selected from pool of 9 remained players means that will be selected any from other remained 7 players = 7/9.
Sought probability means that all three probabilities, that we've already found, appeared at the same time = 9/11 * 8/10 * 7/9 = 28/55
