Respuesta :
Number combination can be used to find the number of selecting n items from m items
The known parameters are;
Number of days the six friends play a game of tennis = Every day = 7 days
Number of weeks the six friends the friends play for = 7 weeks
Therefore the total number of days = 7 × 7 days = 49 days
Required:
(a) The number of ways the friends can play with each other once is given as follows;
[tex]The \ number \ of \ 2 \ combinations \ in \ 6 \ options, C_6^2 = \dfrac{6!}{2!\times (6-2)!} = 15[/tex]
The number of ways the six friends can play with each other once is 15 ways
Therefore, minimum number of singles games that will be between the same friends playing each other is given as follows;
[tex]\mathbf{Number \ of \ games = \dfrac{7 \times 7}{C_6^2} }=3.2\overline 6[/tex]
Therefore, more than 3 of the games or four games will be between pairs playing the game more than four times together, and when at least four of the games of singles are counted it will be between friends playing each other for the fourth time
(b) The number of ways the friends can play game of doubles is given as follows;
[tex]C_6^4 = \dfrac{6!}{4! \times (6 - 4)!} = 15[/tex]
The number of pairs = ₆C₂ = 15
The number of games of doubles = 49
In 49 games, we have;
The number of times the same opposing will play a game of doubles where a member of the pair can be on either side of the court will given as follows;
[tex]\mathbf{Number \ of \ games \ of \ doubles \ between \ same \ pair} = \dfrac{7 \times 7}{15 } =3.2\overline 6[/tex]
Therefore, at least four of the of the games of doubles will be between the same set of four friends playing for the 4th time, which gives; at least four games of doubles between the same set of four friends, two of which are playing (as a pair) the same opposing pair for the 4/2 = second time
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