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Proving Pascal's Triangle and Hockey-Stick Identity using Combinatorics [duplicate]
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This question already has an answer here:
Proof of the Hockey-Stick Identity: $sumlimits_{t=0}^n binom tk = binom{n+1}{k+1}$
13 answers
Combinatorial proof of summation of $sumlimits_{k = 0}^n {n choose k}^2= {2n choose n}$
6 answers
How would I prove the following using a combinatorial proof?
(a) Show that this identity is in Pascal's Triangle:
$$sum_{k=0}^{n} binom{n}{k}^{2} = binom{2n}{n}, ∀n ∈ mathbb{N}$$
(b) Prove the Hockey-Stick Identity:
$$sum_{k=0}^{m} binom{n+k}{k} = binom{n+m+1}{m}, ∀m, n ∈ mathbb{N} text{ with } m ≤ n$$
combinatorics binomial-coefficients
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
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marked as duplicate by JMoravitz, jvdhooft, flawr, rogerl, Mike Earnest Jan 30 at 0:03
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Proof of the Hockey-Stick Identity: $sumlimits_{t=0}^n binom tk = binom{n+1}{k+1}$
13 answers
Combinatorial proof of summation of $sumlimits_{k = 0}^n {n choose k}^2= {2n choose n}$
6 answers
How would I prove the following using a combinatorial proof?
(a) Show that this identity is in Pascal's Triangle:
$$sum_{k=0}^{n} binom{n}{k}^{2} = binom{2n}{n}, ∀n ∈ mathbb{N}$$
(b) Prove the Hockey-Stick Identity:
$$sum_{k=0}^{m} binom{n+k}{k} = binom{n+m+1}{m}, ∀m, n ∈ mathbb{N} text{ with } m ≤ n$$
combinatorics binomial-coefficients
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
$endgroup$
marked as duplicate by JMoravitz, jvdhooft, flawr, rogerl, Mike Earnest Jan 30 at 0:03
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33
add a comment |
$begingroup$
This question already has an answer here:
Proof of the Hockey-Stick Identity: $sumlimits_{t=0}^n binom tk = binom{n+1}{k+1}$
13 answers
Combinatorial proof of summation of $sumlimits_{k = 0}^n {n choose k}^2= {2n choose n}$
6 answers
How would I prove the following using a combinatorial proof?
(a) Show that this identity is in Pascal's Triangle:
$$sum_{k=0}^{n} binom{n}{k}^{2} = binom{2n}{n}, ∀n ∈ mathbb{N}$$
(b) Prove the Hockey-Stick Identity:
$$sum_{k=0}^{m} binom{n+k}{k} = binom{n+m+1}{m}, ∀m, n ∈ mathbb{N} text{ with } m ≤ n$$
combinatorics binomial-coefficients
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
$endgroup$
This question already has an answer here:
Proof of the Hockey-Stick Identity: $sumlimits_{t=0}^n binom tk = binom{n+1}{k+1}$
13 answers
Combinatorial proof of summation of $sumlimits_{k = 0}^n {n choose k}^2= {2n choose n}$
6 answers
How would I prove the following using a combinatorial proof?
(a) Show that this identity is in Pascal's Triangle:
$$sum_{k=0}^{n} binom{n}{k}^{2} = binom{2n}{n}, ∀n ∈ mathbb{N}$$
(b) Prove the Hockey-Stick Identity:
$$sum_{k=0}^{m} binom{n+k}{k} = binom{n+m+1}{m}, ∀m, n ∈ mathbb{N} text{ with } m ≤ n$$
This question already has an answer here:
Proof of the Hockey-Stick Identity: $sumlimits_{t=0}^n binom tk = binom{n+1}{k+1}$
13 answers
Combinatorial proof of summation of $sumlimits_{k = 0}^n {n choose k}^2= {2n choose n}$
6 answers
combinatorics binomial-coefficients
combinatorics binomial-coefficients
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
asked Jan 29 at 22:27
Kishan PatelKishan Patel
164
164
marked as duplicate by JMoravitz, jvdhooft, flawr, rogerl, Mike Earnest Jan 30 at 0:03
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by JMoravitz, jvdhooft, flawr, rogerl, Mike Earnest Jan 30 at 0:03
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33
add a comment |
$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33
$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33
$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33
add a comment |
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$begingroup$
Link for hockey stick identity. That has several proofs, including some combinatorial proofs. Link for sum of squares of binomial coefficients. For those marking as duplicate, try to have two votes for each so they are both marked.
$endgroup$
– JMoravitz
Jan 29 at 22:33