Question

2. At an audition, there are 100 dancers. 35 dancers performed a ballet routine. 63 dancers performed a tap routine, and some dancers performed both ballet and tap routines. 12 dancers choose neither ballet nor tap. How many dancers performed both ballet and tap routines?

Answer 1

To find the number of dancers who performed both ballet and tap routines, we can use the principle of inclusion-exclusion.

Let:
- \( |B| \) be the number of dancers who performed ballet, which is 35.
- \( |T| \) be the number of dancers who performed tap, which is 63.
- \( |N| \) be the number of dancers who performed neither ballet nor tap, which is 12.
- \( |B \cap T| \) be the number of dancers who performed both ballet and tap routines.
- \( |B \cup T| \) be the number of dancers who performed either ballet or tap (or both).

First, we determine the total number of dancers who performed either ballet or tap (or both) using the total number of dancers and those who performed neither:

\[
|B \cup T| = 100 - |N| = 100 - 12 = 88
\]

Now, according to the principle of inclusion-exclusion, we know:

\[
|B \cup T| = |B| + |T| - |B \cap T|
\]

Substituting the numbers we have:

\[
88 = 35 + 63 - |B \cap T|
\]

Calculating \( 35 + 63 \):

\[
35 + 63 = 98
\]

Now we substitute back into the equation:

\[
88 = 98 - |B \cap T|
\]

To find \( |B \cap T| \), we rearrange the equation:

\[
|B \cap T| = 98 - 88 = 10
\]

Thus, the number of dancers who performed both ballet and tap routines is

\[
\boxed{10}
\]