There are 35 students participating in school competitions. Each student must take at least one of these three competitions. 20 students participate in Cricket, 15 students partici... There are 35 students participating in school competitions. Each student must take at least one of these three competitions. 20 students participate in Cricket, 15 students participate in Basketball, and 13 students participate in Football. If 9 students participate in at least two competitions, then what is the number of students who participate in all the competitions? Read more

Steps to Solve

1. Identify Known Values

Let:

• $|C| = 20$ (students in Cricket)
• $|B| = 15$ (students in Basketball)
• $|F| = 13$ (students in Football)
• $|U| = 35$ (total students)
• $|A| = 9$ (students in at least two competitions)

2. Apply the Principle of Inclusion-Exclusion

The principle of inclusion-exclusion for three sets states:

$$ |C \cup B \cup F| = |C| + |B| + |F| - |C \cap B| - |B \cap F| - |F \cap C| + |C \cap B \cap F| $$

Given that $|C \cup B \cup F| = |U|$, we have:

$$ 35 = 20 + 15 + 13 - |C \cap B| - |B \cap F| - |F \cap C| + |C \cap B \cap F| $$

3. Define Variables for Intersections

Let:

• $x = |C \cap B|$
• $y = |B \cap F|$
• $z = |F \cap C|$
• $w = |C \cap B \cap F|$

We also know that the students participating in at least two competitions is:

$$ |A| = x + y + z - 2w $$

Since $|A| = 9$, we can substitute this into our equations.

4. Set-Up the Equations

From the inclusion-exclusion equation:

$$ 35 = 48 - (x + y + z) + w $$

And from the definition of $|A|$:

$$ 9 = x + y + z - 2w $$

5. Solve the System of Equations

Now we can rearrange the equations. From the first equation, rearranging gives:

$$ x + y + z = 48 - 35 + w $$ $$ x + y + z = 13 + w $$

Substituting into the second equation:

$$ 9 = (13 + w) - 2w $$ $$ 9 = 13 - w $$ $$ w = 4 $$

6. Calculate the Number of Students in All Competitions

Substitute $w = 4$ back into one of the equations for $x + y + z$:

$$ x + y + z = 13 + 4 = 17 $$

Now use $x + y + z$ in the equation for $|A|$:

$$ 9 = 17 - 2(4) $$ Which is satisfied and confirms $w = 4$.