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Question

The number of 4digit numbers which are neither multiple of 7 nor multiple of 3 is

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Solution

Let A= set of all four digit integers divisible by 7.
and B= set of all four digit integers divisible by 3.
Total number of 4 digits numbers =9×10×10×10=9000
Let n(A)=x
1001+(x1)7=9996
x=1286

Let n(B)=y
1002+(y1)3=9999
y=3000

Let n(AB)=z
1008+(z1)21=9996
z=429

n(AB)=n(A)+n(B)n(AB) =1286+3000429=3857
n(¯¯¯¯¯¯¯¯¯¯¯¯¯¯AB)=90003857=5143

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n(A∪B) = n(A) + n(B) − n(A∩B)
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