a. 24

b. 66

c. 120

d. 132

**Answer:** b. 66

**Explanation:**

There are 12 celebrities. A handshake needs 2 people.

This simply means in how many ways 2 people can be selected out of 12.

So the answer is ^{12}C_{2}

^{n}C_{r} = | n! |

r!(n-r)! |

∴ ^{12}C_{2} = | 12! | = | 12 x 11 | = 66 = Number of handshakes |

2!(12-2)! | 2 |

Tip:

If there are n people and they shake hands only once with each other, then,

If there are n people and they shake hands only once with each other, then,

Number of handshakes = ^{n}C_{2} = | n(n-1) |

2 |

a. 890

b. 1680

c. 5040

d. 10080

**Answer:** b. 1680

**Explanation:**

We need 6 digits

**So total numbers possible =** 8 x 7 x 6 x 5 = 56 x 30 **= 1680**

a. 27

b. 55

c. 90

d. 144

**Answer:** b. 55

**Explanation:**

We need to SELECT people.

Tip:

Also , **AND = MULTIPLY** ; **OR = ADD**

SELECT = Combination = ^{n}C_{r} = | n! |

r!(n-r)! |

SELECT and ARRANGE = Permutation = ^{n}P_{r} = | n! |

(n-r)! |

3 coaches and 3 batsman or 2 bowlers and 4 coaches means

(3 coaches x 3 batsman) + (2 bowlers x 4 coaches)

Select 3 coaches out of 5 = ^{5}C_{3} = | 5! | = 10 |

3!2! |

Select 3 batsman out of 4 = ^{4}C_{3} = | 4! | = 4 |

3!1! |

Select 2 bowlers out of 3 = ^{3}C_{2} = | 3! | = 3 |

2!1! |

Select 4 coaches out of 5 = ^{5}C_{4} = | 5! | = 5 |

4!1! |

a. 72

b. 190

c. 380

d. 760

**Answer:** c. 380

**Explanation:**

There are 20 stations. Ticket is needed between 2 stops.

That means, we simply need to select 2 stops from possible 20 stops.

Tip:

SELECT = Combination = ^{n}C_{r} = | n! |

r!(n-r)! |

SELECT and ARRANGE = Permutation = ^{n}P_{r} = | n! |

(n-r)! |

That can be done by ^{20}C_{2} ways = | 20! | = | 20! | = 190 ways |

2!(20-2)! | 2!18! |

When we travel from the other side we will need separate ticket.

That means while going from A to B and B to A, we will need separate tickets.

So again from other side we need 190 tickets.

a. 9! X 8!

b.

c.

d. None of the above

**Answer:** b. ^{19}C_{10} x 9! X 8!

**Explanation:**

Here, we first have to select 10 ladies from 19.

Tip:

SELECT = Combination = ^{n}C_{r} = | n! |

r!(n-r)! |

SELECT and ARRANGE = Permutation = ^{n}P_{r} = | n! |

(n-r)! |

Select = Combination

∴ Select 10 ladies =

Tip:

We can arrange 'n' things in 'n!' ways.

But if they are to be arranged in a circle,

then we can arrange them in (n-1)! ways.

Arrange 10 ladies in circle = 10 - 1 = 9! ways

19 - 10 = 9 ladies remain.

Arrange remaining 9 ladies in another circle = 9 - 1 = 8! ways