Let N be the number of ways of choosing a subset of 5 distinct numbers from the set 10 a+b: 1 a 5,1 b 5 where a , b are integers, such that no two of the selected numbers have the same units digit and no two have the same tens digit. What is the remainder when N is divided by 73 ?

Step-by-step solution

10 a+b ; 1 a 5,1 b 5 Let us divide numbers into different sets, such as Set 1= 11,12,13,14,15 Set 2= 21,22,23,24,25 Set 3= 31,32,33,34,35 Set 4= 41,42,43,44,45 Set 5= 51,52,53,54,55 Now to make a number having no two digits and no two ten's digit same, we can select any 1 number from each of set 1 , set 2 , set 3 , set 4 , set 5 . aligned & No. of ways = ^5 C₁ ^4 C₁ ^3 C₁ ^2 C₁ ^1 C₁=120 & 120 73 & Remainder =47 aligned