HAS TO BE DONE BY TOMORROW - A positive integer is called snakelike if its digits alternately increase and decrease or vice versa. For example, 130 and 91727 are snakelike. However 123 is not snakelike (2 is greater than 1 but not 3) and 91727 is not snakelike (7 is greater than 1 but not 7)

A) How many 4-digit snakelike numbers contain each of the digit 1,2,3,4 exactly once

B) How many 3-Digit snakelike numbers have 6 as their middle digit

C) a 7-digit snakelike number starts with digit 9. Explain why it last digit cannot be 0

A) Numbers of the form low, high, ... can start with digits 1 or 2, and the 3rd digit will be the other one. There 4 such numbers 1423, 1324, 2413, 2314. In addition, For a first digit of 2, the second digit may be 1, allowing 2143. Thus, there are 5 numbers starting with 1 or 2 and ending with 3 or 4.

There are 5 additional numbers that are the reverse of these.

There are 10 4-digit snake-like numbers containing 1–4 exactly once.

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B) We assume a "3-digit number" will be in the range 100–999. That is, the leading digit will not be zero. If you allow a leading 0, the answer is different.

With 6 as a middle digit, the numbers 1–5 can be the first digit and the numbers 0–5 can be the last digit. Hence there are 5·6 = 30 numbers of the form low-6-low.

If the form is high-6-high, there are 3 possible first digits and 3 possible last digits, so 3·3 = 9 numbers of the form high-6-high.

There are 39 3-digit snake-like numbers with 6 as the middle digit.

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C) A 7-digit snake-like number will alternate high-low, with high being the last digit. Hence the last digit must be higher than the previous digit, so cannot be 0.