In a class there are 40 students. 16 passed in Chemistry, 20 passed in Physics, 25 passed in Mathematics. 15 students passed in both Mathematics and Physics. 15 students passed in both Mathematics and Chemistry and 10 students passed in both Physics and Chemistry. Find the maximum number of students that passed in all the subjects.

n (C) = 16, n (P) = 20, n (M) = 25

n (MÇP) = n (MÇC) = 15, n (PÇC) = 10,

n (MÇCÇP) = x.

n (CÈPÈM) £ n (U) = 40

n (CÈPÈM) = n (C) + n (P) + n (M) – n (C? M) – n (PÈM) – n (CÇP) + n (CÇPÇM)

40 ³ 16 + 20 + 25 – 15 – 15 – 10 + x

40 ³ 61 – 40 + x

19 ³ x

So maximum number of students that passed all the exams is 19.

<p><span lang="EN-IN">n (C) = 16, n (P) = 20, n (M) = 25</span></p><p><span lang="EN-IN">n (M</span><span lang="EN-IN">Ç</span><span lang="EN-IN">P) = n (M</span><span lang="EN-IN">Ç</span><span lang="EN-IN">C) = 15, n (P</span><span lang="EN-IN">Ç</span><span lang="EN-IN">C) = 10, </span></p><p><span lang="EN-IN">n (M</span><span lang="EN-IN">Ç</span><span lang="EN-IN">C</span><span lang="EN-IN">Ç</span><span lang="EN-IN">P) = x.</span></p><div class="photo-widget-full"><div class="figure"><picture><source srcset="" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/1759128088php6uDjqU.jpeg" alt="" width="436" height="379" loading="lazy"></picture></div></div><p><span lang="EN-IN">n (C</span><span lang="EN-IN">È</span><span lang="EN-IN">P</span><span lang="EN-IN">È</span><span lang="EN-IN">M) </span><span lang="EN-IN">£</span><span lang="EN-IN"> n (U) = 40</span></p><p><span lang="EN-IN">n (C</span><span lang="EN-IN">È</span><span lang="EN-IN">P</span><span lang="EN-IN">È</span><span lang="EN-IN">M) = n (C) + n (P) + n (M) – n (C? M) – n (P</span><span lang="EN-IN">È</span><span lang="EN-IN">M) – n (C</span><span lang="EN-IN">Ç</span><span lang="EN-IN">P) + n (C</span><span lang="EN-IN">Ç</span><span lang="EN-IN">P</span><span lang="EN-IN">Ç</span><span lang="EN-IN">M)</span></p><p><span lang="EN-IN">40&nbsp;</span><span lang="EN-IN">³</span><span lang="EN-IN"> 16 + 20 + 25 – 15 – 15 – 10 + x</span></p><p><span lang="EN-IN">40&nbsp;</span><span lang="EN-IN">³</span><span lang="EN-IN"> 61 – 40 + x</span></p><p><span lang="EN-IN">19&nbsp;</span><span lang="EN-IN">³</span><span lang="EN-IN"> x</span></p><p><span lang="EN-IN">So maximum number of students that passed all the exams is 19.</span></p>

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