Of the 27 participants in a talent show, 13 are singers, 14 are musicians, and 9 are dancers. If 4 people are both singers and dancers, 3 are both singers and musicians and 1 person is all three, how many people are both dancers and musicians

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.

since proposition states that a number is greater than 25, if is true then the number must actually be greater than 25.

this implies that all possible options, except 24, can be accepted

answer

the correct answer is

explanation

the given function is

the excluded values are obtained by solving:

we factor to obtain:

the excluded values are