If m denotes the number of 5-digit numbers such that their digits are in strictly descending order of magnitude and n is the corresponding figure when the digits are in strictly ascending order of magnitude. Then, (m-n) has the value?
Question
If m denotes the number of 5-digit numbers such that their digits are in strictly descending order of magnitude and n is the corresponding figure when the digits are in strictly ascending order of magnitude. Then, (m-n) has the value?
Answer
Let a,b,c,d,e be digits such that a<b<c<d<e. Then all numbers of the form edcba belong to the first category and all numbers of the form abcde belong to the second category. Observe that if a>0, all the numbers appearing in first category will also appear in reverse order in the second category. In our subtraction of (m-n) , all such numbers will cancel each other.
However, complication arises when a=0. Notice that edcba will be a valid 5-digit number but abcde will become a 4-digit number. Hence, (m-n) equals total number of all numbers of the form edcb0 such that 10>e>d>c>b>0. This part is easy to solve. 4 non-zero distinct digits can be chosen in a total of C(9,4) ways. In every such choice, there is exactly one way in which we can arrange these digits in descending order. So, answer = C(9,4) = 126.
Hence, (m-n) = 126.
If m denotes the number of 5-digit numbers such that their digits are in strictly descending order of magnitude and n is the corresponding figure when the digits are in strictly ascending order of magnitude. Then, (m-n) has the value?
Answer
Let a,b,c,d,e be digits such that a<b<c<d<e. Then all numbers of the form edcba belong to the first category and all numbers of the form abcde belong to the second category. Observe that if a>0, all the numbers appearing in first category will also appear in reverse order in the second category. In our subtraction of (m-n) , all such numbers will cancel each other.
However, complication arises when a=0. Notice that edcba will be a valid 5-digit number but abcde will become a 4-digit number. Hence, (m-n) equals total number of all numbers of the form edcb0 such that 10>e>d>c>b>0. This part is easy to solve. 4 non-zero distinct digits can be chosen in a total of C(9,4) ways. In every such choice, there is exactly one way in which we can arrange these digits in descending order. So, answer = C(9,4) = 126.
Hence, (m-n) = 126.
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