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#Combinatorics proof
tacit scaffold
gloomy hawk
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Consider n objects and we choose k of them
Now label one of these objects as "special". Either the "special" object is one of those k in which case we have n-1 remaining objects and need to choose k-1 more. Otherwise we don't choose the "special" object and we have n-1 remaining objects and have to choose k
Combinatorical proof >>>> algebraic proof
(this is called pascal's identity btw)
quaint acorn
Consider n objects and we choose k of them
Now label one of these objects as "s...
Could not wrap my head around this the first time I saw it but it’s a cold proof
Here’s mine it’s the same as arizus but a little more explaining
hidden zinc
❤️ will read in a sec
unreal gyro
Consider n objects and we choose k of them
Now label one of these objects as "s...
this is like that trin q
subsets of {1,2,3,...,n} either involving 1 or not
hidden zinc
this is like that trin q
what question?
unreal gyro
what question?
unreal gyro
hoqw do you do part b here?
if you want to do it along the same lines as the first part you consider subsets involving 1,2, both, or neither
unreal gyro
and then also wb this?
this u can induct probably
or
binomial expansion of (1+1)^n
hidden zinc
if you want to do it along the same lines as the first part you consider subsets...
ah I see
ty
hidden zinc
binomial expansion of (1+1)^n
pls explain a bit more
unreal gyro
well (1+1)^n = nC1 + nC2 + ...
gloomy hawk
how tf u figure that out 💀
Once u've seen it once u can't unsee it 🤷♂️
hidden zinc
Once u've seen it once u can't unsee it 🤷♂️
fair lol
quaint acorn
Sum of pascal triangle rows >
gloomy hawk
how tf u figure that out 💀
Btw the combinatorical proof is by considering how many subsets there are of {1,2,3,...,n}
Which u can count in 2 ways
hidden zinc
Sum of pascal triangle rows >
😮
hidden zinc
Sum of pascal triangle rows >
wait, why is this significant?
quaint acorn
wait, why is this significant?
Sum of each pascal row is double the previous one
You can use your original questions proof to prove that
quaint acorn
(this is called pascal's identity btw)
^
1+3 = 4
Using nCr
I cba to type it rn
hidden zinc
yeah I get how its true, but how does the original proof prove that?
quaint acorn
I guess u can look at it like this, n-1Ck + n-1Ck-1 is the two numbers of Pascal’s triangle from the previous row adding to make the number in the next row (1+3)
There fore the next row after the third one is 1+1+3+3+3+3+1+1
The previous row had 2 3s, the next one is adding 4 3s
Same with 1s