Question

What is the number of projects in which Gyani alone is involved?

• A. Uniquely equal to zero.
• B. Uniquely equal to 1.
• C. Uniquely equal to 4.
• D. Cannot be determined uniquely.
• A. Uniquely equal to zero.
• B. Uniquely equal to 1.
• C. Uniquely equal to 4.
• D. Cannot be determined uniquely.

Answer 1

The Correct Option is D

Step 1: Understand the situation:
Three consultants can be involved in projects in 7 different ways, like the 7 regions of a three-circle Venn diagram. Let G, M and B stand for Gyani, Medha and Buddhi.
Let g, m and b be the projects done by Gyani alone, Medha alone and Buddhi alone.

Step 2: Fill in the regions that are given:
All three together: 6 projects.
Gyani with Medha is 14 projects. This count includes the 6 where Buddhi is also there, so G and M only is \(14 - 6 = 8\).
M and B only: 2. G and B only: 3.
Projects with more than one consultant: \(8 + 2 + 3 + 6 = 19\).

Step 3: Use the total number of projects:
Total projects is one less than twice the number with more than one consultant.
\[ \text{Total} = 2 \times 19 - 1 = 37 \]
Total is also (alone-only projects) + 19, so \(g + m + b = 37 - 19 = 18\).

Step 4: Use the condition on Medha:
"Handled by Gyani and Buddhi individually" means the projects each did on his own, so \(g + b\). This equals all the projects in which Medha is involved.
Medha's total is \(m + 8 + 2 + 6 = m + 16\).
\[ g + b = m + 16 \]

Step 5: Solve:
From the total, \(g + b = 18 - m\). Put this in the line above:
\[ 18 - m = m + 16 \Rightarrow 2m = 2 \Rightarrow m = 1 \]
Then \(g + b = 17\). Only the sum of g and b is fixed.

Step 6: Check whether g is fixed:
Take \(g = 0, b = 17\): total is \(0 + 1 + 17 + 19 = 37\) and \(g + b = 17 = 1 + 16\), so all conditions hold.
Take \(g = 1, b = 16\): total is \(1 + 1 + 16 + 19 = 37\) and \(g + b = 17\), so this also holds.
So g can take many values. Gyani alone cannot be found uniquely.

Step 7: Why the other options are wrong:
Options 1, 2 and 3 say g is uniquely 0, 1 or 4. Each of these values is possible, but so are others, and none is forced. So these options fail.

Final Answer:
The number of projects with Gyani alone cannot be determined uniquely, which is option 4. \[ \boxed{4} \]

Answer 2

Step 1: Understand the situation:
We use the same seven-region Venn set up as in the previous question. Let m be the number of projects in which Medha is involved alone, and let g and b be the solo projects of Gyani and Buddhi.

Step 2: Fill in the shared regions:
All three: 6. Gyani and Medha only: \(14 - 6 = 8\) because the 14 includes the 6 triple projects. Medha and Buddhi only: 2. Gyani and Buddhi only: 3.
More than one consultant: \(6 + 8 + 2 + 3 = 19\).

Step 3: Find the total and the solo projects:
Total \(= 2 \times 19 - 1 = 37\).
Solo projects \(g + m + b = 37 - 19 = 18\).

Step 4: Use the condition on Medha:
Medha's total is \(m + 8 + 2 + 6 = m + 16\). This equals \(g + b\).
Substitute \(g + b = 18 - m\):
\[ 18 - m = m + 16 \]
\[ 2m = 2 \Rightarrow m = 1 \]

Step 5: Why the other options are wrong:
The calculation gives exactly one possible value for m, so it can be determined uniquely. This removes option 4.
That value is 1, not 0 and not 4, so options 1 and 3 are wrong.

Final Answer:
Medha alone is involved in exactly 1 project, which is option 2. \[ \boxed{2} \]