Each student at State University has a student I.D. number consisting of four digits (the first digit is nonzero, and digits may be repeated) followed by three of the letters A, B, C, D, and E (letters may not be repeated). How many different student numbers are possible? Solution Given, I.D. number consists of four digits followed by three letters. In general, it would be of the form, ______________ Let us name the digits as D1, D2, D3 and D4 respectively. And the letters as L1, L2 and L3 respectively. So we have the form, D1D2D3D4L1L2L3 D1 can be any one of 1, 2, 3, 4, 5, 6, 7, 8, 9 (since its given, first digit as non zero) So, there are 9 ways to select the first digit. D2, D3 and D4 can be any one of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (Since its given, digits may be repeated) So, there are 10 ways each to select the next three digits. L1 can be any one of A, B, C, D, E. So, there are 5 ways to select the first letter. And hence, there would be 4 ways to select the second letter L2 (Since its given, letters may not be repeated) Similarly, there would be 3 ways to select the third letter L3 (Since its given, letters may not be repeated) So, the possible number of ways for, D1D2D3D4L1L2L3 would be 9 x 10 x 10 x 10 x 5 x 4 x 3 = 540000 Hence, 540000 different student numbers are possible..