ECS 271 Homework Assignment #7 (Due June 8 2004)
1. If a test contains 20 true-false questions, in how many different ways can a student
mark her test?
2. In how many ways can six persons be seated at a round table?
3. The number of combinations of r objects selected from a set of n objects is written as
nCr. Often it is necessary to use the fact that nCr = nC(n-r). First justify this formula
informally. Then prove the formula mathematically.
4. In how many ways can a subcommittee of 4 persons be chosen from a committee of 10
persons if the chairman of the full committee is required to be on the subcommittee?
5. Consider a random binary sequence such as [0 0 1 0 1 ... 1 0] of length i. Suppose you
wish to generate another random sequence of the same length by tossing a coin i times.
What is the probability that both the strings match exactly?
6. Consider a population A comprised of n of random binary strings, each string of
length i. Consider a test string T, of the same length.
(a) what is the probability that none of the strings in A match T?
(b) what is the probability that at least one string in A matches T?
7. Show that summation n
k
2k
k=0
n
∑=3n.
8. Consider bit strings of length l.
(a) How many possible bit strings of length l are there? ______
(b) Number of possible subsets of bit strings of length l are ______
(c) How many schemas of length l are there? _____
(d) A given bit string of length l is an instance of ____ schemas.
9. Prove that any string of length l is an instance of different schema. ( The best way to
prove this is by illustration. )
10. Consider a ternary string S, of length l composed of the alphabet {0, 1, *}. Assume
that i of the l characters in the string are either zero or one. Stated differently, there are i
fixed positions.
(a) Over these i fixed positions, how many schemata are there?
(b) Over the length l, how many sets of fixed positions are there?