help with combinatorial proofs

pls i need someone to help prove these.

\(proof\) that:

1) nCr=n−1Cr+n−1Cr−1^{n}C_{r} = ^{n-1}C_{r} +^{n-1}C_{r-1}.

2) ∑r=0kmCr+nCk−r=m+nCk\sum {^{k}_{r=0}} ^{m}C_{r} + ^{n}C_{k-r} =^{m+n}C_{k}.

i'll be grateful if anyone can help with these!!!

Note by Samuel Ayinde
6 years, 2 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list
  • bulleted
  • list

1. numbered
2. list
  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai−1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 ∑i=13 \sum_{i=1}^3
\sin \theta sin⁡θ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

1) n−1Cr+n−1Cr−1^{n-1}C_{r}+^{n-1}C_{r-1}

=(n−1)!r!(n−r−1)!+(n−1)!(r−1)!(n−r)!=\frac{(n-1)!}{r!(n-r-1)!}+\frac{(n-1)!}{(r-1)!(n-r)!}

=(n−1)!(r−1)!(n−r−1)!(1r+1n−r)=\frac{(n-1)!}{(r-1)!(n-r-1)!}\left(\frac{1}{r}+\frac{1}{n-r}\right)

=(n−1)!(r−1)!(n−r−1)!(nr(n−r))=\frac{(n-1)!}{(r-1)!(n-r-1)!}\left(\frac{n}{r(n-r)}\right)

=n!r!(n−r)!=nCr=\frac{n!}{r!(n-r)!}=^{n}C_{r}

Omkar Kulkarni - 6 years, 1 month ago

2) Make use of this: ∑r=0knCk−r=∑r=0knCr\displaystyle\sum{_{r=0}^{k}}^{n}C_{k-r}=\displaystyle\sum{_{r=0}^{k}}^{n}C_{r}

I can't seem to find a solution. Do reply if you manage to prove it!

Omkar Kulkarni - 6 years, 1 month ago
×

Problem Loading...

Note Loading...

Set Loading...