Among employee of a company taking vacations last years, $$90$$% took vacations in the summer, $$65$$% in the winter, $$10$$% in the spring, $$7$$% in the autumn, $$55$$% inwinter and summer, $$8$$% in the spring and summer, $$6$$% in the autumn and summer, $$4$$% in the winter and spring, $$4$$% in winter and autumn, $$3$$% in the spring and autumn, $$3$$% in the summer, winter and spring, $$3$$% in the summer, winter and autumn, $$2$$% in the summer, autumn and spring, and $$2$$% in the winter, spring and autumn. Percentage of employee that took vacations during every season:

Question

Among employee of a company taking vacations last years, $$90$$% took vacations in the summer, $$65$$% in the winter, $$10$$% in the spring, $$7$$% in the autumn, $$55$$% inwinter and summer, $$8$$% in the spring and summer, $$6$$% in the autumn and summer, $$4$$% in the winter and spring, $$4$$% in winter and autumn, $$3$$% in the spring and autumn, $$3$$% in the summer, winter and spring, $$3$$% in the summer, winter and autumn, $$2$$% in the summer, autumn and spring, and $$2$$% in the winter, spring and autumn. Percentage of employee that took vacations during every season:

Infinity Learn · Accepted Answer

Suppose that number of employee taking vacations is $$100$$.Su – set of employee taking leave in SummerW – set of employee taking leave in WinterSp – set of employee taking leave in SpringA – set of employee taking leave in $$Autumnn(Su)=90$$,$$n(W)=65$$,$$n(Sp)=10$$,$$n(A)=7n(W$$∩$$Su)=55$$,$$n(Sp$$∩$$Su)=8$$,$$n(A$$∩$$Su)=6n(W$$∩$$Sp)=4$$,$$n(W$$∩$$Au)=4$$,$$n(Sp$$∩$$A)=3n(Su$$∩$$A)=3$$,$$n(Su$$∩W∩$$A)=3n(Su$$∩W∩$$Sp)=3$$,$$n(Su$$∩A∩$$Sp)=2n(W$$∩Sp∩$$A)=2n(Su$$∩Sp∩W∩$$A)=n(Su)+n(Sp)+n(W)+n(A)$$−$$n(Su$$∩$$Sp)$$−$$n(Sp$$∩$$W)$$−$$n(W$$∩$$A)$$−$$n(Su$$∩$$A)$$−$$n(Su$$∩$$W)$$ −$$n(Sp$$∩$$A)+n(Su$$∩Sp∩$$W)+n(Su$$∩W∩$$A)+n(W$$∩A∩$$Su)+n(Su$$∩Sp∩$$A)$$−$$n(Sp$$∪Su∪A∪$$W)$$  $$=90+65+10+7$$−$$55$$−$$8$$−$$6$$−$$4$$−$$4$$−$$3+3+3+2+2$$−$$100=2$$

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