1
GATE CSE 2008
+2
-0.6
Let $${x_n}$$ denote the number of binary strings of length $$n$$ that contains no consecutive $$0s$$.

Which of the following recurrences does $${x_n}$$ satisfy?

A
$${x_n} = 2{x_{n - 1}}$$
B
$${x_n} = {x_{\left[ {n/2} \right]}} + 1$$
C
$${x_n} = {x_{\left[ {n/2} \right]}} + n$$
D
$${x_n} = {x_{n - 1}} + {x_{n - 2}}$$
2
GATE CSE 2008
+2
-0.6
The exponent of $$11$$ in the prime factorization of $$300!$$ is
A
$$27$$
B
$$28$$
C
$$29$$
D
$$30$$
3
GATE CSE 2008
+2
-0.6
In how many ways can $$b$$ blue balls and $$r$$ red balls be distributed in $$n$$ distinct boxes?
A
$${{\left( {n + b - 1} \right)!\left( {n + r - 1} \right)!} \over {\left( {n - 1} \right)!b!\left( {n - 1} \right)!r!}}$$
B
$${{\left( {n + \left( {b + r} \right) - 1} \right)!} \over {\left( {n - 1} \right)!\left( {n - 1} \right)!\left( {b + r} \right)!}}$$
C
$${{n!} \over {b!r!}}$$
D
$${{\left( {n + \left( {b + r} \right) - 1} \right)!} \over {n!\left( {b + r - 1} \right)!}}$$
4
GATE CSE 2008
+2
-0.6
When $$n = {2^{2k}}$$ for some $$k \ge 0$$, the recurrence relation $$T\left( n \right) = \sqrt 2 T\left( {n/2} \right) + \sqrt n ,\,\,T\left( 1 \right) = 1$$\$
evaluates to
A
$$\sqrt n \left( {\log \,n + 1} \right)$$
B
$$\sqrt n \,\log \,n$$
C
$$\sqrt n \,\log \,\sqrt n$$
D
$$n\,\log \sqrt n$$
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