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Chapter 5: Problem 350

The sum of all possible numbers greater than 10000 formed by using the digits from \(\\{1,3,5,7,9\\}\) is (a) 666600 (b) 666660 (c) 66666600 (d) none of these

Short Answer

Expert verified
The sum of all possible numbers greater than 10000 formed by using the digits from the set {1, 3, 5, 7, 9} is 13332000, which is not listed in the given options. Therefore, the correct answer is (d) none of these.

Step by step solution

01

Determine the number of permutations

To find the number of all possible permutations, we will use the fact that the first digit (left-most) can take any of the 5 given digits, the second digit can take any of the remaining 4 digits, the third digit can take any of the remaining 3 digits, and so on. Hence, the number of permutations is given by: \(P = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 5!\). Now, we can say for every 4-digit permutation, there must be a corresponding 5-digit permutation which is obtained by adding another odd digit to the original permutation. So, the total number of permutations considering both 4-digit and 5-digit permutations is: \(P_{total} = 2 \cdot P = 2 \cdot 5! = 2 \cdot 120 = 240\).
02

Determine the sum of all permutations

Now, to find the sum of all permutations, we will use the property of permutation sums which states that when forming numbers of length n with m given digits and finding their sum, each of the m digits will occur with equal frequency at each position. So, for our problem, we have \(n = 5, m = 5\), and our given digits are \(\\{1, 3, 5, 7, 9\\}\). There are \(P_{total} = 240\) permutations. So, the sum of all these permutations (call it S) can be calculated as follows: \[S = (1+3+5+7+9)\left[\frac{240}{5}(1+10+10^2 + 10^3 + 10^4)\right]\]
03

Evaluate the sum

Evaluate the expression for S: \[S = (1+3+5+7+9)\left[\frac{240}{5}(1+10+10^2 + 10^3 + 10^4)\right] = 25\left[\frac{240}{5}(1+10+10^2 + 10^3 + 10^4)\right]\] \[S = 25\cdot(48)\left(1+10+100+1000+10000\right) = 25\cdot48\cdot(11111)\] \[S = 25\cdot 48\cdot 11111 = 13332000\] Since 13332000 is not listed in the given options, the correct answer is: (d) none of these

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