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Lecture on Counting II Warm-up Exercise 1. 1. Flip a fair coin n times. How many possibly outcomes? (2n ) n 2. How many outcomes contain exactly 2 Head? ( 2 ) 3. How many contain at most 2 Heads? ( n0 + n1 + n2 ) 4 How many contain an even number of Heads? Write Pascal triangle, and do the expansion of (x + y)n for n = 0, 1, 2, 3, Terms appearing in (x + y)n : xi y n−i . n Coefficient: i Why? – can be proved by induction. But look at the expansion of (x1 + y1 )(x2 + y2 )(x3 + y3 ). Theorem 1 (Binomial theorem). (x + y)n = n X n n−j j x y . j j=0 Application I: find coefficients Exercise 2. Expansion of (x + y)4 = x4 + 4x3 y + 6x2 y 2 + 4xy 3 + y 4 . Exercise 3. coefficient of x12 y 13 in (x + y)25 . How about in (x + y)27 ? in (2x + 3y)25 ? in (3x − y/2)25 ? Exercise 4. coefficient of x12 in (x + 1)25 , (x − 4)50 , etc. Question: what is the total sum of all coefficients in (2x + 3y)100 ? Application II: evaluation Exercise 5. n X n k k=0 Exercise 6. n X = 2n . n (−1) = (1 + (−1))n = 0. k k=0 n n n n = 2n−1 . Hence 0 + 2 + · · · + 2i + · · · = 2n−1 , and sumk 2k−1 k Exercise 7. n X 2k k=0 Challenging: n X n = 3n . k n 3n + 1 2 = k 2 k k=0,even Application III: identities Exercise 8. Exercise 9. n+1 k+1 n n = + . k k+1 X n X n k k x = x( xk )0 = x((x + 1)n )0 = nx(x + 1)n−1 . k k k k 1