mathmath333
  • mathmath333
Counting question
Mathematics
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mathmath333
  • mathmath333
Counting question
Mathematics
katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{A squad consists of 16 cricketers including Sachin and Rahul. }\hspace{.33em}\\~\\ & \normalsize \text{ In how many ways can the team of 11 cricketers be selected such that,}\hspace{.33em}\\~\\ & \normalsize \text{If Sachin is selected then Rahul is dropped and }\hspace{.33em}\\~\\ & \normalsize \text{if Rahul is selected then Sachin is dropped .}\hspace{.33em}\\~\\ \end{align}}\)
mathmath333
  • mathmath333
amilapsn
  • amilapsn
There are three possible kinds of situations can you guess them:

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mathmath333
  • mathmath333
there are 2 possiblities
amilapsn
  • amilapsn
Question can be interpreted in many ways... ok.. tell me what your two possibilities are...
mathmath333
  • mathmath333
Is S is there R is not vice versa
mathstudent55
  • mathstudent55
Since you can only make a team without one or without the other or without both, think of making a team of 11 out of 15 (no Sachin) and a team of 11 out of 15 (no Rahul).
amilapsn
  • amilapsn
then what about both of them are not in the team?
ganeshie8
  • ganeshie8
isn't the answer simply \[\binom{16}{11} - \binom{14}{11}\]
mathstudent55
  • mathstudent55
\(2(_{15}C_{11})\)
amilapsn
  • amilapsn
@ganeshie8 yes indeed that's another way of doing it...
amilapsn
  • amilapsn
It's much simpler than my method...:)
amilapsn
  • amilapsn
@mathmath333 did you get ganeshie's one?
mathmath333
  • mathmath333
u mean this one \(\Large \dbinom{16}{11} - \dbinom{14}{9}\)
mathmath333
  • mathmath333
not really
amilapsn
  • amilapsn
yep
ganeshie8
  • ganeshie8
it is 14 choose 11 not 14 choose 9
mathmath333
  • mathmath333
but in book it is given this \(\Large \dbinom{16}{11} - \dbinom{14}{9}\)
mathmath333
  • mathmath333
any way i didn't get how that comes
amilapsn
  • amilapsn
it is \(\color{red}{16}\) choose 11 not 14 choose 9
mathmath333
  • mathmath333
ok but how did u get this answer \(\dbinom{16}{11} - \dbinom{14}{11}\)
amilapsn
  • amilapsn
\( \dbinom{16}{11}\) means number of all teams that can be made without considering the condition.. ok?
mathmath333
  • mathmath333
yes
amilapsn
  • amilapsn
Then we just have to remove the number of teams where there are S and R both , right?
ganeshie8
  • ganeshie8
Ahh right, @amilapsn thanks for catching :)
mathmath333
  • mathmath333
ok
amilapsn
  • amilapsn
Then we are left with: 1.teams where there is not S or R 2.teams with only S 3 teams with only R Condition fulfilled...
mathmath333
  • mathmath333
Then we just have to remove the number of teams where there are S and R both this will be done by \(-\dbinom{14}{11}\) ?
amilapsn
  • amilapsn
Yes. Because We've already chosen dravid and thendulkar so we only have to choose anothe nine players out of the rest(14).. Got that?
mathmath333
  • mathmath333
and by other method can u tell me 1.teams where there is not S or R 2.teams with only S 3 teams with only R
amilapsn
  • amilapsn
1-we've choose 11 out of 14 2-we've already chosen S so we have to make sure that R is no there so now we have to choose 10 out of 12 3-the same as 2 we've to choose 10 out of 12
mathmath333
  • mathmath333
2-we've already chosen S so we have to make sure that R is no there so now we have to choose 10 out of \(\color{red}{12}\) 3-the same as 2 we've to choose 10 out of \(\color{red}{12}\) how does \(\color{red}{12}\) comes
amilapsn
  • amilapsn
Let's consider the second situation... After choosing S how much players we are left with?
mathmath333
  • mathmath333
15
amilapsn
  • amilapsn
then can we get R to the team?
mathmath333
  • mathmath333
no
amilapsn
  • amilapsn
Then how much we've to consider?
mathmath333
  • mathmath333
14
amilapsn
  • amilapsn
So we've to choose 10 out of 14... (omg my bad that 12 should be 14... sorry for that mistake)
mathmath333
  • mathmath333
then by the same logic wolfram gives this false http://www.wolframalpha.com/input/?i=is+%5Cbinom%7B16%7D%7B11%7D-%5Cbinom%7B14%7D%7B11%7D%3D%5Cbinom%7B14%7D%7B10%7D%2B%5Cbinom%7B14%7D%7B10%7D%2B%5Cbinom%7B14%7D%7B11%7D+%3F
mathmath333
  • mathmath333
while this is given true http://www.wolframalpha.com/input/?i=is+%5Cbinom%7B16%7D%7B11%7D-%5Cbinom%7B14%7D%7B9%7D%3D%5Cbinom%7B14%7D%7B10%7D%2B%5Cbinom%7B14%7D%7B10%7D%2B%5Cbinom%7B14%7D%7B11%7D+%3F
amilapsn
  • amilapsn
wait a sec... Let me see...
amilapsn
  • amilapsn
Ok so what's the problem? We are alright, know?
mathmath333
  • mathmath333
\(\dbinom{16}{11}-\dbinom{14}{11}\) u said earlier that this is correct but wolfram isn't matching it with the other way
amilapsn
  • amilapsn
that second 11 should be nine: I've said that:\[\text{Yes. Because We've already chosen dravid and thendulkar so we only have to choose}\] \[\text{anothe nine players out of the rest(14).. Got that?}\]
mathmath333
  • mathmath333
ok lol i missed that comment earlier completely thanks
amilapsn
  • amilapsn
Yw!
ganeshie8
  • ganeshie8
hows the test between india and lanka going
amilapsn
  • amilapsn
We won the 1st one you know... It was a legendary match due to Chandimal's playing.. Today match is also a very important for all Sri Lankans because it is the final match of Sanga. We'll miss him so much...
mathmath333
  • mathmath333
In the question why we didn't include the case both sachin and rahul are selected.
amilapsn
  • amilapsn
Because the it's said in the question:\[\large \color{black}{\begin{align} & \normalsize \text{A squad consists of 16 cricketers including Sachin and Rahul. }\hspace{.33em}\\~\\ & \normalsize \text{ In how many ways can the team of 11 cricketers be selected such that,}\hspace{.33em}\\~\\ & \normalsize \text{If Sachin is selected then Rahul is dropped and }\hspace{.33em}\\~\\ & \normalsize \text{if Rahul is selected then Sachin is dropped .}\hspace{.33em}\\~\\ \end{align}}\]
mathmath333
  • mathmath333
by the way india lost 1st test against lanka
mathmath333
  • mathmath333
are u from lanka
amilapsn
  • amilapsn
yep.
mathmath333
  • mathmath333
but it was also not stated that both sachin and rahul won't be selected, then also we add it
amilapsn
  • amilapsn
But it doesn't break the condition right? We have to account for all situations without breaking the condition..
mathmath333
  • mathmath333
but y it dont break the condition
dan815
  • dan815
TotalWays - Ways of sachin and Rahul
dan815
  • dan815
when you to 16 Choose 11 right now in all those combinations it includes the one where both of them are selected that is the only part we have to subtract
amilapsn
  • amilapsn
That's the nature of "if". Have you done logic @mathmath333 ?
mathmath333
  • mathmath333
no
dan815
  • dan815
so 16 Choose 11 - 14 Choose 9
dan815
  • dan815
there are a couple other ways of adding the different cases, this is one way
dan815
  • dan815
another way is to add the cases of no sachin and rahul then only sachin then only rahul
amilapsn
  • amilapsn
\[\href{ https://en.wikipedia.org/wiki/Material_conditional{\huge p\Rightarrow q}\]
mathmath333
  • mathmath333
ok thanks i will look at this logic thing and interpretation of "if"
amilapsn
  • amilapsn
It's pretty simple you'll get it easily.
dan815
  • dan815
2*(14 Choose10) + 14 choose 11 ^ ^________no rahul and sachil '------ only rahul + only sachin also should work
dan815
  • dan815
you can actually go on to show this actually always true 2*(14 Choose10) + 14 choose 11 = 16 choose 11 - 14 choose 9 2*(n choose k) + n choose (k+1) = (n+2) choose (k+1) - n choose(k-1)
dan815
  • dan815
if u like that sorta thing lol.. try proving that
dan815
  • dan815
|dw:1440177672216:dw|

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