Generate largest number with given x active and y inactive bits
The problem at hand involves generating the largest possible number with a specific count of active (set) and
inactive (unset) bits. We are given two integers, x
and y
, representing the counts of
active and inactive bits, respectively. The goal is to find the largest number that satisfies these bit conditions.
Problem Statement and Description
Given integers x
and y
, representing the counts of active and inactive bits respectively,
the task is to find the largest number that satisfies these bit conditions. In other words, we want to create a
binary number with x
active bits followed by y
inactive bits, such that the number is as
large as possible.
Example
For example, if x
is 5 and y
is 2, we want to create the largest possible number that has
5 active bits followed by 2 inactive bits. In binary representation, this number would be 1111100
,
which is equivalent to 124 in decimal. Therefore, the output for this example should be "Largest Number is: 124".

For
x = 5
andy = 2
: The largest possible number with 5 active bits and 2 inactive bits in binary is
1111100
, which is 124 in decimal.  The output is "Given x: 5, y: 2\nLargest Number is: 124".
 The largest possible number with 5 active bits and 2 inactive bits in binary is

For
x = 1
andy = 2
: The largest possible number with 1 active bit and 2 inactive bits in binary is
100
, which is 4 in decimal.  The output is "Given x: 1, y: 2\nLargest Number is: 4".
 The largest possible number with 1 active bit and 2 inactive bits in binary is

For
x = 4
andy = 4
: The largest possible number with 4 active bits and 4 inactive bits in binary is
11110000
, which is 240 in decimal.  The output is "Given x: 4, y: 4\nLargest Number is: 240".
 The largest possible number with 4 active bits and 4 inactive bits in binary is
Idea to Solve the Problem
The core idea to solve this problem is to construct the largest possible number using bitwise operations. We'll
utilize the property of bitwise XOR to set specific bits in the number. To generate a binary number with
x
active bits and y
inactive bits, we'll perform the following steps:
 Calculate
result
as follows:((1 << (x + y))  1)
generates a number withx + y
active bits. This covers the firstx
positions.((1 << y)  1)
generates a number withy
active bits. This covers the lasty
positions. XOR the two results together to get a number with the desired count of active and inactive bits.
Pseudocode
Here's the pseudocode for the algorithm:
function largestNum(x, y):
if x < 0 or y < 0 or (x + y) > 31:
// x and y not in valid range
return
result = ((1 << (x + y))  1) ^ ((1 << y)  1)
print "Given x:", x, ", y:", y
print "Largest Number is:", result
Algorithm Explanation
 The function
largestNum(x, y)
takes two integersx
andy
as inputs.  It first checks whether
x
andy
are within valid ranges. If not, the function returns without processing.  The function calculates
result
as explained earlier.  It prints the values of
x
andy
to show the input.  Finally, the function prints the calculated
result
, which is the largest number with the specified counts of active and inactive bits.
Code Solution

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Time Complexity
The time complexity of this algorithm is constant, as it involves basic arithmetic and bitwise operations that
execute in constant time regardless of the input values. The complexity is not dependent on the magnitude of
x
and y
.
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