Page 144 - 'Blast_Into_Math

Blast into Math! Mathematical perspectives: all aour mase are melong to us

Now, similarly

(n − k)!

is just (n − k − 1)! with one more layer

(n − k)! =(n − k)(n − k − 1)!.

So we can put the multiplicative identity 1 in disguise to re-write

n! n!
+
k!(n − k)! (k +1)!(n − k − 1)!

n! k +1 n! n − k
= +
k!(n − k)! k +1 (k +1)!(n − k − 1)! n − k

(k +1)n! (n − k)n!
= +
(k +1)!(n − k)! (k +1)!(n − k)!

which now can be written in one fraction as

(k +1)n!+ (n − k)n! kn!+ n!+ nn! − kn! (n +1)n! (n +1)!
= = = .
(k +1)!(n − k)! (k +1)!(n − k)! (k +1)!(n − k)! (k +1)!(n − k)!

This means that

n! n! (n +1)!
b
a k+1 n−k + a k+1 n−k = a k+1 n−k .
b
b
k!(n − k)! (k +1)!(n − k − 1)! (k +1)!(n − k)!

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