# Permutations: Variants

## 13 thoughts on “Permutations: Variants”

1. Aniruddh |

Dear Sir,

For the chessboard, why are we not following the symmetry as in the case of circular table, square table, etc?
For example, if the first queen is placed on a black square in row 1, there will be a black square in row 8 from where she’ll have the same view. This way she should have 32 choices and not 64.
Please shed some light on this.

2. Aniruddh, this is pretty much a convention as the squares on a standard chessboard are always labelled (A1 to H8) and hence are considered distinct. I guess most mathematicians also have a fair knowledge of chess so… ðŸ™‚

regards
J

3. Sakshi |

just a cross check,
If in a circular arrangement or any other figure if it is mentioned that all seats are distinct, say, of red, blue, black etc .. then for 10 ppl and 12 seats answer is 12P10 ..right?

4. Sakshi, that’s absolutely right.

regards
J

5. jaspreet |

sir had it been same colored queen,it wd hv been 64C3 for 1st,lekin 2nd wale mein dbt h : diff colors h,toh pehle kisko select iske 3 ways ni hne chahiye : 3C2? matlab 3C2*64*49*36… hw wd the ans differ for same color then??

• I didn’t quite get what you want to ask. Let me reiterate though – if there are three queens of different colours, it is 64 * 49 * 36, while if they are of the same colour it will be (64 * 49 * 36) / 3!

The order in which we place the queens does not matter….if I put a white queen on a1 and then a black queen on c2, it is the same as if I put a black queen on c2 first and then a white queen on a1. Only the final positions matter.

regards
J

6. jaspreet |

yes sir,my bad! gt it nw,thanx ðŸ™‚

7. If suppose question would have asked “no two are in same, row, column and diagonal”. Then how can we approach the problem. (In Diagonal squares are not constant)

• Then it would be a lot nastier – even for 2 queens and not 3, the effort would be quite large; one would have to consider different types of squares and make cases. By the same token, though, they would be less likely to ask such a question in a speed-based competitive test; it would not be a test of understanding…

regards
J

• No, Himanshu, because in a standard chessboard each square is labelled (a1, a2, ….till h7, h8) so the 64 positions are distinct.

regards
J

8. Rohit Gupta |

For the 2nd case in chess board, !st queen can go in 64 ways, 2nd one 63-(14 squares covered under 1st queen), 49 ways, now why can’t 3rd go in 62 boxes-(28 from both 1st and 2nd queen) essentially, 34 squares. Where am i going wrong, Sir?

9. Because there are some common in the 14 of 1st queen and 14 of 2nd, as you would realise if (and only if) you stopped to think a bit. The idea of these questions is to provoke thinking.

regards
J