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Consider the following pseudo code. What is the total number of multiplications to be performed?
D = 2
for i = 1 to n do
for i = 1 to n do
for j = i to n do
for k = j + 1 to n do
D = D * 3
- a)Half of the product of the 3 consecutive integers.
- b)One-third of the product of the 3 consecutive integers.
- c)One-sixth of the product of the 3 consecutive integers.
- d)None of the above.
Correct answer is option 'C'. Can you explain this answer?
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Consider the following pseudo code. What is the total number of multip...
Total number of multiplications
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Consider the following pseudo code. What is the total number of multip...
Explanation:
To determine the total number of multiplications to be performed, we need to analyze the given pseudo code and understand how the loops are nested and executed.
Loop Structure:
The given pseudo code consists of three nested loops:
1. Outer Loop:
- Loop variable: i
- Range: 1 to n
2. Middle Loop:
- Loop variable: j
- Range: i to n
3. Inner Loop:
- Loop variable: k
- Range: j+1 to n
Counting the Multiplications:
To count the number of multiplications, we need to determine how many times the multiplication operation is performed within each nested loop.
1. Outer Loop (i):
- The outer loop runs n times.
- No multiplication operation is performed in this loop.
2. Middle Loop (j):
- The middle loop runs n - i + 1 times.
- No multiplication operation is performed in this loop.
3. Inner Loop (k):
- The inner loop runs n - j times.
- The multiplication operation (D = D * 3) is performed in this loop.
Total Multiplications:
To calculate the total number of multiplications, we need to sum up the number of multiplications performed in each nested loop.
1. Outer Loop:
- Number of multiplications = 0
2. Middle Loop:
- Number of multiplications = 0
3. Inner Loop:
- Number of multiplications = (n - j) * (n - i + 1)
- Summing up the number of multiplications for each iteration of the inner loop:
- For j = i, the number of multiplications = (n - i + 1) * (n - i)
- For j = i + 1, the number of multiplications = (n - i + 1) * (n - i - 1)
- ...
- For j = n, the number of multiplications = (n - i + 1) * (n - n)
- Total number of multiplications in the inner loop = summation of the above values for j = i to n
Calculating the Total:
To calculate the total number of multiplications, we need to sum up the number of multiplications in the inner loop for each iteration of the middle loop. Then, we need to sum up the results for each iteration of the outer loop.
Mathematical Notation:
Let's use the symbol M to represent the total number of multiplications.
M = summation of ((n - j) * (n - i + 1)) for j = i to n for i = 1 to n
Simplifying the Notation:
We can simplify the summation expression by expanding the terms and rearranging them.
M = summation of ((n - j) * (n - i + 1)) for j = i to n for i = 1 to n
= summation of (n * (n - i + 1) - j * (n - i + 1)) for j = i to n for i = 1 to n
= summation of (n * (n - i + 1))
To determine the total number of multiplications to be performed, we need to analyze the given pseudo code and understand how the loops are nested and executed.
Loop Structure:
The given pseudo code consists of three nested loops:
1. Outer Loop:
- Loop variable: i
- Range: 1 to n
2. Middle Loop:
- Loop variable: j
- Range: i to n
3. Inner Loop:
- Loop variable: k
- Range: j+1 to n
Counting the Multiplications:
To count the number of multiplications, we need to determine how many times the multiplication operation is performed within each nested loop.
1. Outer Loop (i):
- The outer loop runs n times.
- No multiplication operation is performed in this loop.
2. Middle Loop (j):
- The middle loop runs n - i + 1 times.
- No multiplication operation is performed in this loop.
3. Inner Loop (k):
- The inner loop runs n - j times.
- The multiplication operation (D = D * 3) is performed in this loop.
Total Multiplications:
To calculate the total number of multiplications, we need to sum up the number of multiplications performed in each nested loop.
1. Outer Loop:
- Number of multiplications = 0
2. Middle Loop:
- Number of multiplications = 0
3. Inner Loop:
- Number of multiplications = (n - j) * (n - i + 1)
- Summing up the number of multiplications for each iteration of the inner loop:
- For j = i, the number of multiplications = (n - i + 1) * (n - i)
- For j = i + 1, the number of multiplications = (n - i + 1) * (n - i - 1)
- ...
- For j = n, the number of multiplications = (n - i + 1) * (n - n)
- Total number of multiplications in the inner loop = summation of the above values for j = i to n
Calculating the Total:
To calculate the total number of multiplications, we need to sum up the number of multiplications in the inner loop for each iteration of the middle loop. Then, we need to sum up the results for each iteration of the outer loop.
Mathematical Notation:
Let's use the symbol M to represent the total number of multiplications.
M = summation of ((n - j) * (n - i + 1)) for j = i to n for i = 1 to n
Simplifying the Notation:
We can simplify the summation expression by expanding the terms and rearranging them.
M = summation of ((n - j) * (n - i + 1)) for j = i to n for i = 1 to n
= summation of (n * (n - i + 1) - j * (n - i + 1)) for j = i to n for i = 1 to n
= summation of (n * (n - i + 1))
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Consider the following pseudo code. What is the total number of multiplications to be performed?D = 2for i = 1 to n dofor j = i to n dofor k = j + 1 to n do D = D * 3a)Half of the product of the 3 consecutive integers.b)One-third of the product of the 3 consecutive integers.c)One-sixth of the product of the 3 consecutive integers.d)None of the above.Correct answer is option 'C'. Can you explain this answer?
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