The correct option is A 66−( 17C6−6 11C5)
Let x1,x2⋯x6 be the number that appears on the six dice.
Let us find the number of ways to get the sum less than or equal to 17.
This will be same as finding the number of solutions to the inequality x1+x2+x3+...x6≤17
Introducing a dummy variable x7 (x7≥0), the inequality becomes an equation :
1+x2+x3+...x6+x7=17
Here ,1≤xi≤6 where i=1,2,...6 and x7≥0
Therefore number of solutions
=coeff. of x17 in (x+x2+⋯+x6)6×(1+x+x2+⋯)
=coeff. of x11 in (1−x6)6(1−x)−7
=coeff. of x11 in (1−6x6)(1−x)−7
= 17C6−6× 11C5
Total number of cases =66
Hence, the number of ways to get a sum greater than 17=66−( 17C6−6×11C5)