Math for Computer science
Questions 281 to 290
281.

What is the 5^{th}
term of (3 + x)^{8 }?
(a)
C(8, 4) 3^{4} x^{4} (b) C(8, 5) 3^{3} x^{5} (c)
C(8, 4) 3^{4} x^{5}
(d)
C(8, 4) 3^{5} x^{4} (e) C(7, 4) 3^{4} x^{4}.

How many different
strings can be made by reordering the letters of the word ABRACADABRA
(a) 8 ! / 5!2!2! (b) 11! / 5!2!2! (c) 5!2!2! / 11! (d)
5!2!2! / 8! (e) 11!.


What is the value
of k after the following code has been executed.
k := 0
for i_{1} := 1
to n_{1}
for
i_{2} := 1 to n_{2}
for i_{2} :=
1 to n_{2}
:
:
for
i_{m} := 1 to n_{m}
k := k + 1
(a) n_{1} + n_{2} + n_{3}
+….n_{m} (b) n_{1}.n_{2}.n_{3}…………n_{m }(c) C(n+m1,
m)
(d) C(n,m) (e) P(n,m).


GCD of 2^{4}.3^{5}.7^{2}
and 2^{4}.3^{2 }is
(a) 2^{2}.3^{5}.7^{2} (b) 2^{4}.3^{5}.7^{2} (c) 2^{5}.3^{4}.7^{2} (d) 2^{4}.3^{2}.7^{0} (e) 2^{2}.3^{5}.7^{4}.


How many solutions
are there to the equation x1 + x2 +x3+x4 = 17
(a) C(20,17) (b) C(20 , 3) (c) 1140
(d)
All the above (e) none of the above.


Fermat’s theorem
states that If p is prime and a is an
integer not divisible by p, then
(a) a^{p–1 }_{} 1_{ }(mod p) (b) a^{p }_{} 1_{ }(mod p) (c) a^{p–1 }_{} 0_{ }(mod p)
(d)
a^{p–1 }_{} 1_{ }(mod p–1) (e) a^{p }_{} 1_{ }(mod p–1).


Warshall’s
algorithm is used for finding the __________ of a relation.
(a)
reflexive closure (b) symmetric closure (c) transitive closure
(d)
transpose (e) inverse.


Suppose that A is a
subset if V*. Then the set consisting of concatenations of arbitrarily many
strings from A represented by A* is called
(a) Symmetric closure (b)
Reflexive closure
(c) Kleene closure
(d)
Finite closure (e) Finite state closure.


The finitestate
machine with no output is called _________
(a) Kleene machine (b)
Finite machine
(c) Outputfree machine
(d)
Finitestate automata (e) Moore machine.


Type 2 grammars are
also called __________
(a) Contextsensitive grammars (b) Regular grammars c)
Contextfree grammars
(d)
Context less grammars (e) Irregular grammars.

Answers
281.

Answer : (a)
Reason : For finding the 5^{th} term use the
general term formula t_{r+1 }= C(x, y)x^{n–r }y^{r}.
Here you have to find the term t_{r+1}.

Answer : (b)
Reason : This is a problem of permutations with
repetitions. If n be the total number of letters and r1 letters are of same
type and r2 letters are of same type then the number of different strings by
reordering the letters of the word is n!/r1! × r2!


Answer : (b)
Reason : By the definition of product rule.


Answer : (d)
Reason : Gcd of 2^{4}.3^{5}.7^{2}
and 2^{4}.3^{2 }is 2 ^{min(4,4)} 3 ^{min(5,2) }7
^{min(2,0)} = 2^{4}.3^{2}.7^{0}


Answer : (d)
Reason : This is combinations with indistinguishable
objects. Here n=4 and r=17. The number of solutions is equal to the number of
17combinations with repletion allowed from
a set with 4 elements. Therefore the answer is C(4+171,17) = C(20,17)
= C(20,3).


Answer : (a)
Reason : Definition of Fermat’s theorem


Answer : (c)
Reason : Warshall’s algorithm is one of the algorithms
for finding the transitive closure of the relation.


Answer : (c)
Reason : Definition of the kleene’s closure


Answer : (d)
Reason : Definition of finitestate automata


Answer : (c)
Reason : Type 2 grammars are also called contextfree
grammars.

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