Permutations & Combinations on CHESSBOARD/ GRID.....

Question

Q1. How many squares are there on a 8cm x 8cm chessboard/grid ?

(a) 64 (b) 36 (c) 1296 (d) 102 (e) 204

Q2. How many rectangles are there on a 8 x 8 chessboard/grid ? 
(a) 64 (b) 36 (c) 1296 (d) 102 (e) 204

Q3. In how many ways can two 1cm x 1cm squares be selected in a 8cm x 8cm chessboard/grid such that they have a side common? 
(a) 64 (b) 32 (c) 102 (d) 112 (e) 56

Q4. In how many ways can a 1cm x 1cm black square be selected followed by a 1cm x 1cm white square in a 8cm x 8cm chessboard/grid such that they are not in the same row or same column?

(a) 768 (b) 24 (c) 32 (d) 393 (e) 786

Q5. In how many ways can two 1cm x 1cm black squares be selected in a 8cm x 8cm chessboard/grid such that they are not in the same row or same column?

(a) 24 (b) 16 (c) 32 (d) 25 (e) 400

Q6. How many rectangles are there on a 6 x 6 chessboard/grid ? 
(a) 91 (b) 441 (c) 36 (d) 8281 (e) 1764

Please explain your answers in detail..thanks
1. e 
2. c 
3. d 
4. a 
5. e 
6. b

jumsumtak · Answer

to find number of squares in 8x8 chessboard: (draw the figure with 8 rows and 8 columns)

calculate the number of squares with area 1x1: (easy - all the small squares) - 8x8  
calculate the number of squares with area 2x2: in the first row you will be able to find 7 sides with length 2 and there are 7 such columns as well. So the total number of such squares is 7x7.
|
|
|
the total number of squares will be: 1^2+2^2+3^2+....8^2 
1^2+2^2+...n^2= n(n+1)(2n+1)/6; substitute 8 for n and get the answer.

for the number of rectangles in a nxn square: it is sum of n^3 = 1^3+2^3+...8^3
1^3+2^3+3^3...+n^3= (n(n+1)/2)^2 substitute 8 and get the number of rectangles

similarly, for rest of the questions make cases from the diagram. Post the specific cases that you have made for any one answer you're not getting and someone shall chip in!

pariearth · Answer

Thanks  jumsumtak  I particularly need explaination of 
 Q5. In how many ways can two 1cm x 1cm black squares be selected in a 8cm x 8cm chessboard/grid such that they are not in the same row or same column?

jumsumtak · Answer

you may get to the answer by eliminating options itself. only 400 seems close to the answer (it has to be greater than 32 - because that is the number of black squares on the board. right?)

anyhow, the solution:

number of black squares on the board = 32 (4 in each of the 8 rows - 4 in each of the 8 columns)

ways of selecting 1 black square out of 32 = 32
that makes us exclude 7 black squares for our next selection (3 each from the row and the column we picked our first black square and that selected black square itself)
hence, ways of selecting the second black square = 32-7=25

so total ways = 32x25=800

now because the 2 black squares are identical, you have counted them twice in your calculation. you have counted cases for the same pair where a black box was considered first selection and was considered the second selection separately.

we want selection and not permutation, so divide by 2. you get 400.

did that help?

ConnectTheDots · Answer

Total 32 black squares
4 black squares in each row and each column

1st square can be chosen in 32 ways.
2nd square can be chosen in 25 ways = 32 - (1 chosen + 3 in same row + 3 in same column)
total = 32x25 = 800.
Am I missing something ?

jumsumtak · Answer

Did you get the last para of the previous post? 
Basically, you need to divide by 2 because you have counted each pair twice.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Permutations & Combinations on CHESSBOARD/ GRID.....

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards