SOLUTION: Show that the way in which the entries in Pascal's triangle are formedby adding "above left" and "above right' is consistent with the following statement; nCr-1 +nCr =n+1Cr Us

Algebra ->  Permutations -> SOLUTION: Show that the way in which the entries in Pascal's triangle are formedby adding "above left" and "above right' is consistent with the following statement; nCr-1 +nCr =n+1Cr Us      Log On


   

Question 947191: Show that the way in which the entries in Pascal's triangle are formedby adding "above left" and "above right' is consistent with the following statement;
nCr-1 +nCr =n+1Cr
Use the fact that nCr = n!/(n-r)!r! to show that the above statement is true.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Question 940228
nC(r-1) + nC(r) ?=? (n+1)Cr
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(1)   nCr = n%21%2F%28r%21%28n-r%29%21%29

(2)   nC(r-1) = n%21%2F%28%28r-1%29%21%28n%5E%22%22-%28r-1%29%29%21%29%29 = n%21%2F%28%28r-1%29%21%28n-r%2B1%29%21%29

(3)   (n+1)Cr = %28n%2B1%29%21%2F%28r%21%28%28n%2B1%29%5E%22%22-r%29%21%29%29 = %28n%2B1%29%21%2F%28r%21%28n%2B1-r%29%21%29

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We want to prove that expression(2) + expression(1) = expression(3)

nC(r-1) + nCr =

n%21%2F%28%28r-1%29%21%28n-r%2B1%29%21%29%22%22%2B%22%22n%21%2F%28r%21%28n-r%29%21%29

Substitute (n-r+1)(n-r)! for (n-r+1)! and r(r-1)! for r!

n%21%2F%28%28r-1%29%21%28n-r%2B1%29%5E%22%22%28n-r%29%21%29%22%22%2B%22%22n%21%2F%28r%28r-1%29%21%28n-r%29%21%29

LCD = r(r-1)!(n-r+1)(n-r)!

n%21%2Ar%2F%28%28r-1%29%21%28n-r%2B1%29%28n-r%29%21r%29%22%22%2B%22%22n%21%28n-r%2B1%29%2F%28r%28r-1%29%21%28n-r%29%21%28n-r%2B1%29%21%29

replace r(r-1)! by r! and replace (n-r+1)(n-r) by (n-r+1)!

+n%21r%2F%28+r%21%28n-r%2B1%29%21%29++%22%22%2B%22%22n%21%28n-r%2B1%29%2F%28r%21%28n-r%2B1%29%21%29

%28n%21r%2Bn%21%28n-r%2B1%29%29%2F%28r%21%28n-r%2B1%29%21%29

%28n%21r%2Bn%21n-n%21r%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29

%28cross%28n%21r%29%2Bn%21n-cross%28n%21r%29%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29

%28n%21n%2Bn%21%29%2F%28r%21%28n-r%2B1%29%21%29

%28n%2B1%29n%21%2F%28r%21%28n-r%2B1%29%21%29

%28n%2B1%29%21%2F%28r%21%28n-r%2B1%29%21%29

(n+1)Cr

Edwin