Logical Reasoning Discussion

**Common Information**

At the annual sports meet, 106 sportsmen participated in hockey, 122 in football and 120 in cricket. It is known that 48 participated in hockey and football and 70 in football and cricket. A total of 200 sportsmen participated in the meet.

Q. |
If 10 sportsmen did not participate in any sport and 54 participated in hockey and cricket, how many participated in all three? |

✖ A. |
11 |

✖ B. |
12 |

✖ C. |
13 |

✔ D. |
14 |

**Solution:**

Option(**D**) is correct

To solve this question set, let us introduce a variable *x *that represents the set of players that participated in all three sports. In the Venn diagram, this set will be represented by the area that is common to all three circles. In symbolic form, it means:

n(*H *∩ *F* ∩ *C*) = *x*

Now, it is given that the number of students that participated in both Hockey and Football is 48.

This means that the number of students that participated only in both Football and Hockey and not Cricket = 48 – *x*

Similarly, the number of students that participated in Football and Cricket but not in Hockey = 70 – *x*

In this way, we can label all the sections of the Venn diagram in terms of known numbers and the variable *x*.

After incorporating data from the first question (that is said to be valid for the second question as well), this is the completed Venn diagram:

Now, we can easily solve the question using basic linear equations.

Refering to the Venn diagram

After appropriating cardinal numbers to the various zones on the Venn diagram, we get,* x* + (54 - *x*) + (48 - *x*) + (70 - *x*) + (*x* + 4) + (*x* - 4) + (*x* + 4) = 200 – 10

Simplifying, *x* = 14