Vietnam
A ^@3^@-digit number is of the form ‘high-low-high’ ^@ - ^@ that is the tens digit is smaller than both the hundreds digit and the units (or ‘ones’) digit. How many such ^@3^@-digit numbers are there?
Answer:
^@ 285 ^@
- A ^@3-^@digit number is of the form ‘high-low-high’, so, the tens digit of the ^@3^@-digit number cannot be ^@9,^@ as the units and the hundreds, digit needs to be larger than tens digit and ^@9^@ is the largest digit. Therefore, the smallest tens digit of the ^@3^@-digit number of the required form is ^@ 0 ^@ and the largest tens digit of the ^@3^@-digit number is ^@ 8. ^@
- If the tens digit is ^@ 0, ^@ then the hundreds digit can be any digit from ^@ 1 ^@ to ^@ 9, ^@ and the units digit can also be any digit from ^@ 1 ^@ to ^@ 9. ^@
So, there are ^@ 9 \times 9 ^@ possible numbers of the required form with ^@ 0 ^@ at tens place.
If the tens digit is ^@ 1, ^@ then the hundreds digit can be any digit from ^@ 2 ^@ to ^@ 9, ^@ and the units digit can also be any digit from ^@ 2 ^@ to ^@ 9. ^@
So, there are ^@ 8 \times 8 ^@ possible numbers of the required form with ^@ 1 ^@ at tens place. - Similarly, possible numbers with ^@ 2 ^@ at tens place is ^@ 7 \times 7, ^@ possible numbers with ^@ 3 ^@ at tens place is ^@ 6 \times 6, \ldots , ^@ possible numbers with ^@ 8 ^@ at tens place is ^@ 1 \times 1. ^@
Therefore, the total number of possible ^@3^@-digit numbers of the required form ^@ = (9 \times 9) + (8 \times 8) + \ldots + (1 \times 1) = 285 ^@ - Hence, there are ^@ 285 \space 3^@-digit numbers of the form ‘high-low-high’.
