15-(CESPE)Em um tribunal,todos os 100 técnicos administrativos falam ingles ou espanhol, 55 deles falam ingles e 65 falam espanhol.Nessa situação,quantos técnicos falam inglês e espanhol?

## Step 1
The problem involves the concept of set theory, specifically the principle of inclusion and exclusion. The principle of inclusion and exclusion states that for any two sets, the size of their union is the size of the first set plus the size of the second set minus the size of their intersection.

## Step 2
In this problem, the two sets are the technicians who speak English and the technicians who speak Spanish. The total number of technicians is 100, which is the size of the union of the two sets.

## Step 3
The number of technicians who speak English is 55, and the number of technicians who speak Spanish is 65. These are the sizes of the two sets.

## Step 4
To find the number of technicians who speak both English and Spanish (the intersection of the two sets), we subtract the total number of technicians from the sum of the number of technicians who speak English and the number of technicians who speak Spanish.

### **The formula is:**
### $Number\, of\, technicians\, who\, speak\, both\, English\, and\, Spanish = (Number\, of\, technicians\, who\, speak\, English + Number\, of\, technicians\, who\, speak\, Spanish) - Total\, number\, of\, technicians$

## Step 5
Substituting the given values into the formula, we get:

### $Number\, of\, technicians\, who\, speak\, both\, English\, and\, Spanish = (55 + 65) - 100 = 120 - 100 = 20$