Show that n(n-1 r) = (r+1)(n r+1)
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Step-by-step explanation:
n + 1
r
=
n
r
+
n
r − 1
.
Proof
n
r
+
n
r − 1
=
n!
r!(n − r)! +
n!
(r − 1)!(n − r + 1)!
=
n!
(r − 1)!(n − r)!
1
r
+
1
n − r + 1
=
n!
(r − 1)!(n − r)!
n + 1
r(n − r + 1)
=
(n + 1)!
r!(n − r + 1)! =
n + 1
r
Lemma 1.2 n
r
is an integer, for n, r integers with n ≥ 0, 0 ≤ r ≤ n.
Proof
Let P(n) be the statement that n
r
is an integer for all r with 0 ≤ r ≤ n. We prove this for all n ≥ 0.
(i) If n = 0 then the only value of r with 0 ≤ r ≤ n is r = 0. Now P(0) is true, since
0
0
=
0!
0!0! =
1
1 × 1
=
n + 1
r
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