Question

All 40 persons prefer to use at least one of the exercise activities. 25 prefer yoga, 20 prefer run, 18 prefer cycle, 6 yoga and run, 8 run and cycle, 11 yoga and cycle. How many of the total 40 persons prefer all three exercises? 2

Answer 1

$\\ Let\ \ person\ prefer \ yoga\ is\ represented \ by\ Y\\ person\ prefer \ run \ is\ represented \ by \ R\\ person \ prefer\ cycle \ is\ represented\ by \ C\\ Given\\ n(Y)=25\\ n(R)=20\\ n(C)=18\\\ n(Y\cap R)=6\\ n(R\cap C)=8\\ n(Y\cap C)=11\\ n(Y\cup R\cup C)=40\\ \\ {so ,\ n(Y\cap R\cap C)=n(Y\cup R\cup C)-n(Y)-n(R)-n(C)+n(Y\cap R)+n(R\cap C)+n(Y\cap C)}\\ \Rightarrow n(Y\cap R\cap C)=40-25-20-18+6+8+11\\ \Rightarrow n(Y\cap R\cap C)=2\\ \\ \\ \boldsymbol{So,\ correct\ choice \ is\ 2}$

2 people prefer all three exercises