Page 167 - Fixtures

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n(n − 1)[C(n − 2, 0) + C(n − 2,1) + + C(n − 2,n − 2)]
= n(n − 1)2n−2

[AIME 1991] ( )

(1 + 0.2)1000

(1 + 0.2)1000 = C (1000, 0)(0.2)0 + C(1000, 0.2)1

+ C(1000,2)(0.2)2 + + C (1000,1000)(0.2)1000

= A0 + A1 + A2 + + A1000

.k = 0,1, ,1000 , Ak = C(1000,k)(0.2)k

Ak k

Ak k Ak

, .Ak

 1 k  1 k +1
5 5
C (1000,k) > C (1000, k + 1)

1000! > 1000 !

k !(1000 − k)! 5(k + 1)!(1000 − k − 1)!

1 >1
k !(1000 − k)(1000 − k − 1)! 5(k + 1)k !(1000 − k − 1)!

1 >1
1000 − k 5(k + 1)

5k + 5 > 1000 − k

k > 165.8 k

.166 k

[AIME 1986] ( )

1 − x + x2 − x 3 + + x16 − x17