Question

Counting question

Answer 1

\(\large \color{black}{\begin{align}
& \normalsize \text{A squad consists of 16 cricketers including Sachin and Rahul. }\hspace{.33em}\\~\\
& \normalsize \text{ In how many ways can the team of 11 cricketers be selected such that,}\hspace{.33em}\\~\\
& \normalsize \text{If Sachin is selected then Rahul is dropped and }\hspace{.33em}\\~\\
& \normalsize \text{if Rahul is selected then Sachin is dropped .}\hspace{.33em}\\~\\
\end{align}}\)

Answer 2

@amilapsn

Answer 3

There are three possible kinds of situations can you guess them:

Answer 4

there are 2 possiblities

Answer 5

Question can be interpreted in many ways... ok.. tell me what your two possibilities are...

Answer 6

Is S is there R is not vice versa

Answer 7

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).

Answer 8

then what about both of them are not in the team?

Answer 9

isn't the answer simply \[\binom{16}{11} - \binom{14}{11}\]

Answer 10

\(2(_{15}C_{11})\)

Answer 11

@ganeshie8 yes indeed that's another way of doing it...

Answer 12

It's much simpler than my method...:)

Answer 13

@mathmath333 did you get ganeshie's one?

Answer 14

u mean this one
\(\Large \dbinom{16}{11} - \dbinom{14}{9}\)

Answer 15

not really

Answer 16

yep

Answer 17

it is 14 choose 11

not 14 choose 9

Answer 18

but in book it is given this \(\Large \dbinom{16}{11} - \dbinom{14}{9}\)

Answer 19

any way i didn't get how that comes

Answer 20

it is \(\color{red}{16}\) choose 11

not 14 choose 9

Answer 21

ok but how did u get this answer

\(\dbinom{16}{11} - \dbinom{14}{11}\)

Answer 22

\( \dbinom{16}{11}\) means number of all teams that can be made without considering the condition.. ok?

Answer 23

yes

Answer 24

Then we just have to remove the number of teams where there are S and R both , right?

Answer 25

Ahh right, @amilapsn thanks for catching :)

Answer 26

ok

Answer 27

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...

Answer 28

Then we just have to remove the number of teams where there are S and R both

this will be done by \(-\dbinom{14}{11}\) ?

Answer 29

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?

Answer 30

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

Answer 31

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

Answer 32

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

Answer 33

Let's consider the second situation... After choosing S how much players we are left with?

Answer 34

15

Answer 35

then can we get R to the team?

Answer 36

no

Answer 37

Then how much we've to consider?

Answer 38

14

Answer 39

So we've to choose 10 out of 14... (omg my bad that 12 should be 14... sorry for that mistake)

Answer 40

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

Answer 41

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

Answer 42

wait a sec... Let me see...

Answer 43

Ok so what's the problem? We are alright, know?

Answer 44

\(\dbinom{16}{11}-\dbinom{14}{11}\)
u said earlier that this is correct but wolfram isn't matching it with the other way

Answer 45

that second 11 should be nine:
I've said that:\[\text{Yes. Because We've already chosen dravid and thendulkar so we only have to choose}\]
\[\text{anothe nine players out of the rest(14).. Got that?}\]

Answer 46

ok lol i missed that comment earlier completely thanks

Answer 47

Yw!

Answer 48

hows the test between india and lanka going

Answer 49

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...

Answer 50

In the question why we didn't include the case both sachin and rahul are selected.

Answer 51

Because the it's said in the question:\[\large \color{black}{\begin{align}
& \normalsize \text{A squad consists of 16 cricketers including Sachin and Rahul. }\hspace{.33em}\\~\\
& \normalsize \text{ In how many ways can the team of 11 cricketers be selected such that,}\hspace{.33em}\\~\\
& \normalsize \text{If Sachin is selected then Rahul is dropped and }\hspace{.33em}\\~\\
& \normalsize \text{if Rahul is selected then Sachin is dropped .}\hspace{.33em}\\~\\
\end{align}}\]

Answer 52

by the way india lost 1st test against lanka

Answer 53

are u from lanka

Answer 54

yep.

Answer 55

but it was also not stated that both sachin and rahul won't be selected, then also we add it

Answer 56

But it doesn't break the condition right? We have to account for all situations without breaking the condition..

Answer 57

but y it dont break the condition

Answer 58

TotalWays - Ways of sachin and Rahul

Answer 59

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

Answer 60

That's the nature of "if".
Have you done logic @mathmath333 ?

Answer 61

no

Answer 62

so 16 Choose 11 - 14 Choose 9

Answer 63

there are a couple other ways of adding the different cases, this is one way

Answer 64

another way is to add
the cases of no sachin and rahul
then only sachin
then only rahul

Answer 65

\[\href{
https://en.wikipedia.org/wiki/Material_conditional {\huge p\Rightarrow q}\]

Answer 66

ok thanks i will look at this logic thing and interpretation of "if"

Answer 67

It's pretty simple you'll get it easily.

Answer 68

2*(14 Choose10) + 14 choose 11
^ ^________no rahul and sachil
'------ only rahul + only sachin also should work

Answer 69

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)

Answer 70

if u like that sorta thing lol.. try proving that