Out of a class of 100 students, 25 play at least cricket and football, 15 play at least cricket and hockey, 12 play at least football and hockey and 10 play all the three sports. The number of students playing cricket, football and hockey are 50, 37 and 22, respectively. The number of students who do NOT play any of the three sports is<\/p>\n
[amp_mcq option1=”33″ option2=”23″ option3=”27″ option4=”30″ correct=”option1″]<\/p>\n
We need to find the number of students who do NOT play any of the three sports. This is given by Total students – |C \u222a F \u222a H|.
\nWe use the Principle of Inclusion-Exclusion for three sets:
\n|C \u222a F \u222a H| = |C| + |F| + |H| – (|C \u2229 F| + |C \u2229 H| + |F \u2229 H|) + |C \u2229 F \u2229 H|
\nSubstitute the given values:
\n|C \u222a F \u222a H| = 50 + 37 + 22 – (25 + 15 + 12) + 10
\n|C \u222a F \u222a H| = 109 – (52) + 10
\n|C \u222a F \u222a H| = 109 – 52 + 10
\n|C \u222a F \u222a H| = 57 + 10 = 67.<\/p>\n
The number of students who play at least one sport is 67. Out of a class of 100 students, 25 play at least cricket and football, 15 play at least cricket and hockey, 12 play at least football and hockey and 10 play all the three sports. The number of students playing cricket, football and hockey are 50, 37 and 22, respectively. The number of students who … <\/p>\n
\nThe number of students who do NOT play any of the three sports = Total students – |C \u222a F \u222a H|
\nNumber of students who do not play any sport = 100 – 67 = 33.
\n<\/section>\n
\nWe can also visualize this with a Venn diagram. The value 10 goes in the center. The values for exactly two sports are:
\nOnly C and F = |C \u2229 F| – |C \u2229 F \u2229 H| = 25 – 10 = 15
\nOnly C and H = |C \u2229 H| – |C \u2229 F \u2229 H| = 15 – 10 = 5
\nOnly F and H = |F \u2229 H| – |C \u2229 F \u2229 H| = 12 – 10 = 2
\nThe values for exactly one sport are:
\nOnly C = |C| – (Only C&F) – (Only C&H) – (C&F&H) = 50 – 15 – 5 – 10 = 20
\nOnly F = |F| – (Only C&F) – (Only F&H) – (C&F&H) = 37 – 15 – 2 – 10 = 10
\nOnly H = |H| – (Only C&H) – (Only F&H) – (C&F&H) = 22 – 5 – 2 – 10 = 5
\nTotal playing at least one sport = (Only C) + (Only F) + (Only H) + (Only C&F) + (Only C&H) + (Only F&H) + (C&F&H)
\n= 20 + 10 + 5 + 15 + 5 + 2 + 10 = 67.
\nNumber not playing any sport = 100 – 67 = 33. This confirms the result.
\n<\/section>\n","protected":false},"excerpt":{"rendered":"