Answer Key:

OA: 41/44Ways in which the first couple can sit together = 2 * 4! (1 couple is considered one unit)
Ways for second couple = 2 * 4!
These cases include an extra case of both couples sitting together
Ways in which both couple are seated together = 2 * 2 * 3! = 4! (2 couples considered as 2 units- so each couple can be arrange between themselves in 2 ways and the 3 units in 3! Ways)
Thus total ways in which at least one couple is seated together = 2 * 4! + 2 * 4! - 4! = 3 * 4!
Total ways to arrange the 5 ppl = 5!
Thus, prob of at least one couple seated together = 3 * 4! / 5! = 3/5
Thus prob of none seated together = 1 - 3/5 = 2/5OA: 39/49OA: 14/23OA: 1/4OA : 62.5%OA: 1/26OA: 3/8OA: 1/720OA: 1/23/8OA : 0.62Cyclicity for 3^m = 3,9,7,1
Cyclicity for 7^n = 7,9,3,1
Case 1 : 3^m edns with 9* 7^n ends with 1
Case 2 : 3^m ends with 7, and the other ends with 1
Case 3: 3^m ends with 1 and the other ends with 7.......625 * 3/100 * 100 (if m and n can be same as well).
hence OA: 3/16no. of rectangle = 7C2 * 7C2= 441 , no of square = 1+4+9+16+25+36= 91 Ans 91/441when dividing a number by 5, the remainders can be 0,1,2,3,4
means 5 cases
so when one number is div by 5 and rest are not
then 4c1 * 4^3
now all div by 5- 1 case
3 numbers div by 5 - 4c3 * 4
2 numbers div by - 4c2 * 4^2
Now (1 + 4c3 * 4 + 4c2 * 16 + 4c1 * 64)/5^4
= (1+16+96+256)/625
= 369/625OA : 5/8total numbers=4!=24 last digit is 6=3!=6numbers last digit 4=3!=6 numbers 12/24=1/2Consider the first row. There are 7 possibilities. (choosing consecutive squares ) 7 * 8 = 56 (for 8 rows)
Similarly for columns = 56. Hence 56 * 2 =112
Probability : 112/ 64c2 (choosing 2 squares from the board)OA: 7/164C2*D(2)/4! = =1/410 sets(10_11_12.....19)
Each set 9. So 9c2.
10*9c2/90c2 = 8/89HCF=1 and their LCM is their product so zero1/2(10,6,6) : 3 * 1/9 * (1/3)^2 = 3/81 = 27/729
(10,10,6): 3 * (1/9)^2 * (1/3) = 3/243 = 9/729
(10,10,2): 3 * (1/9)^2 * 5/9 = 15/729.
(10,10,10): (1/9)^3 = 1//729
dd: (27+9+15+1)/729 = 52/729Consider 3 consecutive HHH case. :
HHHHH,HHHHT,HHHTH,HHHTT,THHHT,TTHHH,HTHHH,THHHH so total 8 cases. similarly for TTT we get 8 cases. Total cases = 16
P = 16/32= 1/2all three are of form 3k=5c3
all three are of form 3k+1 = 5c3
all three are of form 3k+2 = 5c3
one is 3k,one is 3k+1 and one is 3k+2 = 5c1 * 5c1 * 5c1=125
so total 125+30=155
total cases = 15c3 = 455
probability = 155/455= 31/911a+b+c= 12. so 11C2
6+a+b+c= 12..so 5C2.
so 11C2 - 3 * 5C2 = 55-3 * 10= 25. so 25/216