Answer

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**Hint:**We can solve this problem by using a general substitution method and also using the Venn diagram method. Given in the problem is the information about a number of persons taking two medical tests in a given number of groups of people. We have to find a number of people for the required result by using probability relations. Then, using the formula and given information we can find the number of students who can speak both.

**Formula used:**We will apply the given into the formula of \[n(D \cup B) + n(D \cap B) = n(D) + n(B)\].

Here,

\[D\] means the number of people who have been diagnosed with diabetes and

\[B\] means the number of people who have been diagnosed with blood pressure.

**Complete step-by-step answer:**

It is given that; total number of people diagnosed with is \[50\].

Number of people diagnosed with diabetes is \[30\].

Number of people diagnosed with blood pressure is \[40\].

We have to find the number of people who have been diagnosed with both the diseases.

So, as per the given information

\[n(D \cup B) = 50\]

\[n(D) = 30\]

\[n(B) = 40\]

Let us consider the number people who have been diagnosed with both the diseases is \[x\] that is \[n(D \cap B) = x\].

We have to find the value of \[n(D \cap B)\].

We know that,

\[n(D \cup B) + n(D \cap B) = n(D) + n(B)\]

Substitute the values in the above formula we get,

\[\Rightarrow 50 + x = 40 + 30\]

Simplifying we get,

\[\Rightarrow x = 20\]

Hence, the number of people who have been diagnosed with both the diseases is \[20\].

**$\therefore $ The correct answer is option C**

**Note:**We can solve the sum by using a Venn diagram.

Here, the red shaded part indicates the number of people diagnosed with diabetes is \[30\].

The blue shaded part indicates the number of people diagnosed with blood pressure is \[40\].

The green shaded part indicates the number of people who have been diagnosed with both the diseases.

The total number of people who have been diagnosed with is \[50\].

We have to find the value of the green shaded part.

So, the value of green shaded part is

\[\Rightarrow (40 + 30) - 50 = 20\]

Hence, the number of people who have been diagnosed with both the diseases is \[20\].

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