Of the 27 participants in a talent show, 13 are singers, 14 are musicians, and 9

3 people are both dancers and musicians

Step-by-step explanation:

We solve this problem looking from the center (the person who is all three) to the outside info:

if 1 person is all three, and 4 people are singers and dancers, that means 3 people are JUST singers and dancers

We do the same for the 3 people that are singers and musicians: one of them is all three professions, so 2 of them are JUST singers and musicians

We have 13 singers, and among them 6 have other professions (1 is all three, 3 are dancers too, and 2 are musicians too), so 7 people are JUST singers

let's call X the number of people that are JUST dancers, Y the number of people that are JUST musicians, and Z the number of people that are JUST dancers and musicians (the answer for this problem)

From the 14 musicians, 2 are singers as well, and 1 is all three, so 14-3-Z people are JUST musicians (Y = 11 - Z)

From the 9 dancers, 3 are singers as well, and 1 is all three, so 9-4-Z people are JUST dancers (X = 5 - Z)

if we sum all the people that are JUST some professions, we have the 27 participants, so:

7+3+2+1+X+Y+Z = 27 -> X+Y+Z = 14

using the values of X and Y in function of Z, we have:

5 - Z + 11 - Z + Z = 14 -> Z = 2

2 people are JUST dancers and musicians. With the 1 person that is all three, we have 3 people that are both dancers and musicians.