# Set Theory, Calendars, Clocks and Binomial Theorem

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## Set Theory: Number of Elements

Set A comprises all three digit numbers that are multiples of 5, Set B comprises all three–digit even numbers that are multiples of 3 and Set C comprises all three–digit numbers that are multiples of 4. How many elements are present in ${A\cup B\cup C}$?
1. 420
2. 405
3. 555
4. 480

Choice A. 420

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## Detailed Solution

This question is more of a Number Systems question than a Set Theory one!!

Set A = {100, 105, 110, ….995} ${\mapsto}$ {5 × 20, 5 × 21, ….. 5 × 199} ${\mapsto}$ 180 elements.

Set B = {102, 108, 114, ……996} ${\mapsto}$ {6 × 17, 6 × 18, 6 × 19, ….. 6 × 166} ${\mapsto}$ 150 elements.

Set C = {100, 104, 108, …..996} ${\mapsto}$ {4 × 25, 4 × 26, ….. 4 × 249} ${\mapsto}$ 225 elements.

${A\cap B}$ = {120, 150, 180, …..990} ${\mapsto}$ All 3-digit multiples of 30 ${\mapsto}$ 30 elements.

${ B\cap C}$ = {108, 120, 132, …..996} ${\mapsto}$ All 3-digit multiples of 12 ${\mapsto}$ 75 elements.

${ C\cap A}$ = {120, 140, 160, …..980} ${\mapsto}$ All 3-digit multiples of 20 ${\mapsto}$ 45 elements.

${ A\cap B\cap C}$ = {120, 180, …..960} ${\mapsto}$ All 3-digit multiples of 60 ${\mapsto}$ 15 elements.

${A\cup B\cup C}$ = A + B + C – ${A\cap B}$${ B\cap C}$${C\cap A}$ + ${A\cap B\cap C}$

= 180 + 150 + 225 – 30 – 75 – 45 + 15 = 420 Answer choice (a)

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