All the five-digit numbers in which each succ | vedclass

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Question

(B) five-digit number with strictly increasing digits is uniquely determined by choosing $5$ distinct digits from the set ${1, 2, 3, 4, 5, 6, 7, 8, 9}

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(B) five-digit number with strictly increasing digits is uniquely determined by choosing $5$ distinct digits from the set ${1, 2, 3, 4, 5, 6, 7, 8, 9}$. The digit $0$ cannot be used because the digits must be strictly increasing,and $0$ would have to be the first digit,which is not allowed for a five-digit number.Total such numbers = $^9C_5 = 126$.Numbers starting with $1$: We need to choose $4$ more digits from ${2, 3, 4, 5, 6, 7, 8, 9}$.Number of such numbers = $^8C_4 = 70$.Numbers starting with $2$: We need to choose $4$ more digits from ${3, 4, 5, 6, 7, 8, 9}$.Numbers starting with $23$: We choose $3$ more digits from ${4, 5, 6, 7, 8, 9}$.Number of such numbers = $^6C_3 = 20$.Cumulative count so far = $70 + 20 = 90$.Numbers starting with $24$:Numbers starting with $245$: We choose $2$ more digits from ${6, 7, 8, 9}$.Number of such numbers = $^4C_2 = 6$.Cumulative count = $90 + 6 = 96$.The $97^{th}$ number is the first number starting with $246$. The digits must be strictly increasing,so the next digits must be $7$ and $8$.The $97^{th}$ number is $24678$.This number does not contain the digit $5$.

All the five-digit numbers in which each successive digit exceeds its predecessor are arranged in the increasing order of their magnitude. The $97^{th}$ number in the list does not contain the digit:

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