3. Write a 9-digit number where each digit is bigger than the previous one. How many such numbers exist?
4. Strike out 10 digits from the number 123451234512345 so that the remaining number is as large as possible.
5. The words 'zero' and 'one' share letters 'e' and 'o'. The words 'one' and 'two' share a letter 'o', and the words 'two' and 'three' also share a letter 't'. How far do you have to count to find two consecutive numbers which do not share an English letter in common?
Answer (2)
The 9-digit number with increasing digits is only 123456789. After removing 10 digits from 123451234512345, the largest remaining number is 54321. The first pair of consecutive numbers without shared letters are 7 and 8.
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Let's tackle each part of the question step by step:
Write a 9-digit number where each digit is bigger than the previous one. How many such numbers exist?
To form a 9-digit number where each digit is bigger than the previous digit, you need to use different digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}. Each digit must be chosen uniquely to maintain the strictly increasing order.
The number of such numbers is the number of combinations of 9 different digits taken from a set of 9, which is exactly 1 (since you must use all 9 digits).
Thus, there is exactly 1 such number.
Strike out 10 digits from the number 123451234512345 so that the remaining number is as large as possible.
To make the remaining number as large as possible, we need to keep the biggest digits possible. We start from the left, keeping the biggest numbers when possible:
Look at the sequence 123451234512345 and choose:
Keep the first '5', and then remove as many smaller numbers as necessary.
After removing digits, the largest possible number formed from the given digits could be: 54345 .
How far do you have to count to find two consecutive numbers which do not share an English letter in common?
We analyze the letters of the numbers to determine if they share any common letters:
'zero' and 'one' share 'e' and 'o'.
'one' and 'two' share 'o'.
'two' and 'three' share 't'.
To avoid sharing, continue with:
'four' and 'five': share no common letters.
Therefore, the first pair of consecutive numbers that do not share any letter in their English spelling are 4 ("four") and 5 ("five").
Thus, you need to count up to 5 to find such a pair.
