World Cup fever!
India and South Africa play one day International Series until one team wins 4 matches . No match ends in a draw. Find in how many ways the series can be won.
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The answer is 70.
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5 solutions
For case after second one...you took the total number of wins as the total number of matches. For example, What i'm sayin is that, in second case, instead of 4!/3! , it should be 5!/4! i.e. total number of matches/no.of wins.
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bro i will elaborate the solution more to make it better understandable. and yeah i have used n − 1 C a where a is the number of matches won by the losing team or say the number of matches lost by the winning team. and n is the total number of matches played.
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I get that, but it would be nice if you can derive the formula...thanks, it will help a lot!:)
I think its over-rated.... It should be a level 2 or level 3 problem...
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well just to tell you i aint d one who rates it... it is based on number of solvers and attempts..
Can someone Please help me understand Why in the 2nd ,3rd and 4th cases you have used n-1 C a instead of n C a?
That's a good solution...But next time u upload a sum like this support India....Watch the World Cup....if u know what i mean...
For the series to end, there have to be atmost 7 matches.
Let India be represented by a and South Africa by b
We get the following set
a a a a b b b
The no. of permutations of a and b would be 4 ! 3 ! 7 ! = 3 5
Now taking India as b and South Africa as a we again get 3 5
Thus total 3 5 + 3 5 = 7 0
Hey vaibhav though your answer is correct . However the method is wrong. You have assumed that other team wins 3 matches . But it is not necessary than other team does win 3 matches also suppose if this is the result aaaa and no win for b , then there will only be 4 matches .
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isn't that what the set which I have given shows.
a a a a b b b
This set shows that India has already won 4 matches with SA winning no matches (or vice versa) and the rest of b s are too complete the set
Similarly, the set
a a b a a b b
Shows that India won the series after 5 matches, and the rest 2 b s have to be shown no matter what
Thats right @starwar clone , but then how is is his answer right?? If he skipped the cases??
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What @Vaibhav Prasad has done here (but not explicitly stated in his solution), is to create a bijection between a possible series where A wins, and a sequence of 4 a's and 3 'bs.
This is done by taking a series, and then adding additional b's to the end of it. For example, if the series was abaa, then we form abaabb by adding 2 more b's to it.
(Note that you still have to prove that this is a bijection).
For more detailed explanations, see this problem .
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@Calvin Lin – Calvin Lin , it's funny that I do not know what a bijection is, yet used it.
Anyways, where can I know more about it ??
@Calvin Lin – Okay thanks, @Calvin Lin , and sorry for the inconvenience @Vaibhav Prasad and @Harshvardhan Mehta !!
He means to say that :
The maximum number of matches that can be played is 7. In which there should be 4 wins and 3 loses. So the number of ways in which a team can win the series is the number of permutation of 4 wins and 3 loses. i.e. 4 ! × 3 ! 7 ! = 3 5 Similarly for the other team. We get 7 0 ways.
am i wrong @Vaibhav Prasad
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@Harshvardhan Mehta – Ye, but what i am saying, is that this case considers only the series of 4 wins, and 3 losses. But it has nowhere been mentioned in the question that we need to consider the uttermost case. There have to be the consideration of cases with (4 wins, 2 losses), (4 wins, 1 loss) and (4 wins, no loss).
short and sweet.....upvoted
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Thats what i asked @Harshvardhan Mehta Why there can't be a series of 6 matches in which one team wins 4 and the other 3???
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hey i have already considered that case...
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@Harshvardhan Mehta – then how come, you and @Vighnesh Raut have got the same answer????
hey vaibhav correct answer should be 102. FOR INDIA LLLL no. of combinations =1 LLLWL no. of combinations=5 LLLWWL no. of combinations=15 LLLWWWL no. of combinations=35 SAME FOR SOUTH AFRICA HENCE TOTAL NO. OF COMBINATION=2*(1+5+15+35)=102
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in LLLWL number of combinations is 4 as LLLLW is not taken a case. similarly for all others..
All series may come from any combination of: W.W.W.W.L.L.L.L. = 4 ! × 4 ! 8 ! = 7 0
A more elegant solution would be to find out the number of ways to arrange 4 'W' 's (wins) and 3 'L' 's (loss) in a row. = 7!/(4!3!)=35 (for one team)
If There are 4 matches won, obviously there are at most seven matches. So take the answer of 7C4 and multiply by 2, because both sides can win. The answer is 35x2=70
Taking I for India and S for South Africa.
Suppose South Africa wins the series, then the last match is always won by South Africa.
⇒ Total no. of ways = 35
In the same number of ways India can win the series.
⇒ Total no. of ways in which the series can be won = 3 5 + 3 5 = 7 0