Well done Ang Zhi Ping, age 15, River Valley High School, Singapore for your
excellent solution to this question.
Let the binomial coefficient n!/r!(n−r)! be denoted by
n r
.
(1)
By considering powers of (1 + x) show that
n ∑ k=0
n k
2
=
2n n
As (1 + x)n(1 + x)n = (1 + x)2n (1) , we write down the
Binomial expansion giving:
n ∑ p=0
n p
xp
n ∑ q=0
n q
xq
=
2n ∑ r=0
2n r
xr.
The left hand side of the equation is
n
0
+
n
1
x + …+
n n
xn
n
0
+
n
1
x + …+
n n
xn
.
So the coefficient of xn on the left hand side of (1) is
n
0
n n
+
n
1
n n−1
+
n
2
n n−2
+ …+
n n−1
n
1
+
n n
n
0
.
Since
n r
=
n n−r
(2)
we see that the coefficient of xn on the left hand side of (1) is
n ∑ k=0
n k
2
.
(3)
As the coefficient of xn on the right hand side of (1) is