Quiz 2Discrete Mathematics Course (January-April 2015)

(Submission Deadline: 17th February 2015)

1. (5 + 10 Marks) Let us represent numbers in the decimal expansion. Then

(a) Count the number of n digit integers in which no two consecutive digits are same.(b) Count the number of n digits integers in which no three consecutive digits are same

i. Let T (n) be the number of n digits integers in which no three consecutive digits aresame. Obtain a recurrence relation for T (n).

ii. (Bonus) Solve the recurrence relation.

2. (4 + 4 + 4 Marks) Let T (n) = T (bn/2c) + T (dn/2e) + dn2 e and T (1) = 1. Is T (n)

(a) (n1001/1000),(b) O(n(log logn)8),(c) (1000000n log 5n2),

3. (5+4 Marks) In how many ways can 20 identical balls be distributed among 5 distinct binssuch that every bin has at least one ball? In how many ways can 5 identical balls be distributedamong 20 distinct bins?

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