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If the number of subsets X of {1, 2, 3, …. 10} such that X contains at least two elements and no two elements of X differ by 1 is K, then sum of digits of K is equal to
Correct answer is '7'. Can you explain this answer?
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If the number of subsets X of {1, 2, 3, . 10} such that X contains at ...
Every allowable k-elements subset corresponds to a way of choosing k out of a row of 10 objects so that no two are adjacents, remove (k − 1) unselected objects, one from each gap.
This establishes, for each k ≥ 2 , a one to one correspondence between allowable subsets of {1, 2, …… 10} containing k elements, and the number of ways of choosing k out (10 − k+ 1) objects. It follows that there are
Allowable subsets.
This establishes, for each k ≥ 2 , a one to one correspondence between allowable subsets of {1, 2, …… 10} containing k elements, and the number of ways of choosing k out (10 − k+ 1) objects. It follows that there are
Allowable subsets.
Most Upvoted Answer
If the number of subsets X of {1, 2, 3, . 10} such that X contains at ...
To solve this problem, we need to find the number of subsets of {1, 2, 3, ..., 10} that satisfy two conditions: the subset must contain at least two elements, and no two elements in the subset can differ by 1.
Counting the Subsets:
Let's consider the elements 1, 2, 3, ..., 10 one by one and decide whether to include them in the subset or not. There are two possibilities for each element:
1. Exclude the element: If we exclude an element, we can freely choose or exclude any other element from the remaining set. So, there are 2 possibilities for each element when it is excluded.
2. Include the element: If we include an element, we need to exclude its adjacent elements (i.e., elements that differ from it by 1). Since there are no adjacent elements for 1, we can include it in the subset without any restrictions. However, for the other elements (2 to 10), we can include them only if their adjacent elements are not included in the subset. This means that we need to exclude two elements (the element itself and its adjacent element) for each included element.
So, for each element from 2 to 10, there are 2 possibilities when it is excluded and 1 possibility when it is included. Therefore, the total number of subsets that satisfy the given conditions is:
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 2^9 = 512
Calculating the Sum of Digits:
Now, we need to find the sum of the digits of K, which is 512. Let's calculate it:
5 + 1 + 2 = 8
Therefore, the sum of the digits of K is 8, not 7.
Correcting the Answer:
The correct answer is 8, not 7.
Counting the Subsets:
Let's consider the elements 1, 2, 3, ..., 10 one by one and decide whether to include them in the subset or not. There are two possibilities for each element:
1. Exclude the element: If we exclude an element, we can freely choose or exclude any other element from the remaining set. So, there are 2 possibilities for each element when it is excluded.
2. Include the element: If we include an element, we need to exclude its adjacent elements (i.e., elements that differ from it by 1). Since there are no adjacent elements for 1, we can include it in the subset without any restrictions. However, for the other elements (2 to 10), we can include them only if their adjacent elements are not included in the subset. This means that we need to exclude two elements (the element itself and its adjacent element) for each included element.
So, for each element from 2 to 10, there are 2 possibilities when it is excluded and 1 possibility when it is included. Therefore, the total number of subsets that satisfy the given conditions is:
2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 1 = 2^9 = 512
Calculating the Sum of Digits:
Now, we need to find the sum of the digits of K, which is 512. Let's calculate it:
5 + 1 + 2 = 8
Therefore, the sum of the digits of K is 8, not 7.
Correcting the Answer:
The correct answer is 8, not 7.
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If the number of subsets X of {1, 2, 3, . 10} such that X contains at least two elements and no two elements of X differ by 1 is K, then sum of digits of K is equal toCorrect answer is '7'. Can you explain this answer?
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