How many different three-digit-numbers between 100 and 1,000 have five as the tens digit

Question

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Answer 1

Let the tens digit be 5.

 

The ones digit goes from 0 to 9 and the hundreds digit goes from 1 to 9.  That accounts for 100 to 999.  1000 does not have a five in the tens digit, so it doesn't count.

 

That means 10*9=90 possibilities.  (10 for ones; 9 for hundreds)

 

They are:

 

150, 151, 152, 153, 154, 155, 156, 157, 158, 159,
250, 251, 252, 253, 254, 255, 256, 257, 258, 259,
350, 351, 352, 353, 354, 355, 356, 357, 358, 359,
450, 451, 452, 453, 454, 455, 456, 457, 458, 459,
550, 551, 552, 553, 554, 555, 556, 557, 558, 559,
650, 651, 652, 653, 654, 655, 656, 657, 658, 659,
750, 751, 752, 753, 754, 755, 756, 757, 758, 759,
850, 851, 852, 853, 854, 855, 856, 857, 858, 859,
950, 951, 952, 953, 954, 955, 956, 957, 958, 959

Answer 2

The number can be represented as d₁*100 + 50 + d_2, where d₁ and d₂ are digits.

 

Now d₁ can have values between 1 and 9, and there are 9 possible values.

Now d₂ can have values between 0 and 10, and there are 10 possible values.

 

The total number of different three digit numbers between 100 and 1000 that have five as the tens digit is d₁*d₂ = 90