Lesson Remarkable identities for Binomial Coefficients
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<H2>Remarkable identities for Binomial Coefficients</H2> <H3>Problem 1</H3>Prove this identity for <B>Binomial Coefficients</B> {{{C[n]^0}}} + {{{C[n]^1}}} + {{{C[n]^2}}} + . . . + {{{C[n]^k}}} + . . . + {{{C[n]^n}}} = {{{2^n}}} for any positive integer number <B>n</B>. <B>Solution</B> The proof is very straightforward. Simply substitute {{{x}}} = {{{1}}} into the <B>Binomial Expansion</B> {{{(1 + x)^n}}} = = {{{ sum ( C[n]^i*x^i , i=0, n ) }}} (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Permutations/Binomial-Theorem.lesson?content_action=edit_dev>Binomial Theorem</A> under the current topic in this site). You will get the required identity. This is how it looks for the low order binomial coefficients: <B>n</B> = 1, {{{C[1]^0}}} = 1, {{{C[1]^1}}} = 1, {{{C[1]^0}}} + {{{C[1]^1}}} = 1 + 1 = 2; <B>n</B> = 2, {{{C[2]^0}}} = 1, {{{C[2]^1}}} = 2, {{{C[2]^2}}} = 1, {{{C[2]^0}}} + {{{C[2]^1}}} + {{{C[2]^2}}} = 1 + 2 + 1 = 4; <B>n</B> = 3, {{{C[3]^0}}} = 1, {{{C[3]^1}}} = 3, {{{C[3]^2}}} = 3, {{{C[3]^3}}} = 1, {{{C[3]^0}}} + {{{C[3]^1}}} + {{{C[3]^2}}} + {{{C[3]^3}}} = 1 + 3 + 3 + 1 = 8. <H3>Problem 2</H3>Prove this identity for <B>Binomial Coefficients</B> {{{C[n]^0}}} - {{{C[n]^1}}} + {{{C[n]^2}}} + . . . + {{{(-1)^k*C[n]^k}}} + . . . + {{{(-1)^n*C[n]^n}}} = {{{0}}} (alternate sum) for any positive integer number <B>n</B>. <B>Solution</B> Again, the proof is very straightforward. Simply substitute {{{x}}} = {{{-1}}} into the <B>Binomial Expansion</B> {{{(1 + x)^n}}} = = {{{ sum ( C[n]^i*x^i , i=0, n ) }}} (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Permutations/Binomial-Theorem.lesson?content_action=edit_dev>Binomial Theorem</A> under the current topic in this site). You will get the required identity. This is how it looks for the low order binomial coefficients: <B>n</B> = 1, {{{C[1]^0}}} = 1, {{{C[1]^1}}} = 1, {{{C[1]^0}}} - {{{C[1]^1}}} = 1 - 1 = 0; <B>n</B> = 2, {{{C[2]^0}}} = 1, {{{C[2]^1}}} = 2, {{{C[2]^2}}} = 1, {{{C[2]^0}}} - {{{C[2]^1}}} + {{{C[2]^2}}} = 1 - 2 + 1 = 0; <B>n</B> = 3, {{{C[3]^0}}} = 1, {{{C[3]^1}}} = 3, {{{C[3]^2}}} = 3, {{{C[3]^3}}} = 1, {{{C[3]^0}}} - {{{C[3]^1}}} + {{{C[3]^2}}} - {{{C[3]^3}}} = 1 - 3 + 3 - 1 = 0. <H3>Problem 3</H3>Prove this identity for <B>Binomial Coefficients</B> {{{C[n]^0}}} + {{{2*C[n]^1}}} + {{{2^2*C[n]^2}}} + . . . + {{{2^k*C[n]^k}}} + . . . + {{{2^n*C[n]^n}}} = {{{3^n}}} for any positive integer number <B>n</B>. <B>Solution</B> Similar to the <B>Problem 1</B> and <B>Problem 2</B> above, the proof is very straightforward. Simply substitute {{{x}}} = {{{2}}} into the <B>Binomial Expansion</B> {{{(1 + x)^n}}} = = {{{ sum ( C[n]^i*x^i , i=0, n ) }}} (see the lesson <A HREF=http://www.algebra.com/algebra/homework/Permutations/Binomial-Theorem.lesson?content_action=edit_dev>Binomial Theorem</A> under the current topic in this site). You will get the required identity. By doing in a similar way, you can prove many other identities for Binomial Coefficients, like these {{{C[n]^0}}} + {{{3*C[n]^1}}} + {{{3^2*C[n]^2}}} + . . . + {{{3^k*C[n]^k}}} + . . . + {{{3^n*C[n]^n}}} = {{{4^n}}}, {{{C[n]^0}}} + {{{(1/2)*C[n]^1}}} + {{{(1/2^2)*C[n]^2}}} + . . . + {{{(1/2^k)*C[n]^k}}} + . . . + {{{(1/2^n)*C[n]^n}}} = {{{(3/2)^n}}}, {{{C[n]^0}}} - {{{(1/2)*C[n]^1}}} + {{{(1/2^2)*C[n]^2}}} + . . . + {{{(((-1)^k)/2^k)*C[n]^k}}} + . . . + {{{(((-1)^n)/2^n)*C[n]^n}}} = {{{1/2^n}}}. For the close subject on the <B>Pascal's triangle</B> see the lesson <A HREF=http://www.algebra.com/algebra/homework/Permutations/The-Pascal-triangle.lesson>The Pascal's triangle</A> under the current topic in this site. My lessons on Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion in this site are - <A HREF =http://www.algebra.com/algebra/homework/Permutations/Binomial-Theorem.lesson>Binomial Theorem, Binomial Formula, Binomial Coefficients and Binomial Expansion</A> - Remarkable identities for Binomial Coefficients (this lesson) - <A HREF =http://www.algebra.com/algebra/homework/Permutations/The-Pascal-triangle.lesson>The Pascal's triangle</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Solved-problems-on-binomial-coefficients.lesson>Solved problems on binomial coefficients</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Solving-equations-that-include-binom-coefts-and-numbers-of-permutations.lesson>Solving equations that include binomial coefficients and numbers of permutations</A> - <A HREF=https://www.algebra.com/algebra/homework/Permutations/Math-circle-level-problem-on-binomial-coefficients.lesson>Math circle level problem on binomial coefficients</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-of-lessons-on-Binomial-Expansion-Binomial-coefficients-and-Pascal%27s-triangle.lesson>OVERVIEW of lessons on Binomial Expansion, Binomial coefficients and the Pascal's triangle</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
