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The number of increasing function from f...

The number of increasing function from `f : AtoB` where `A in {a_(1),a_(2),a_(3),a_(4),a_(5),a_(6)}`, `B in {1,2,3,….,9}` such that `a_(i+1) gt a_(i) AA I in N` and `a_(i) ne i` is

A

`30`

B

`28`

C

`24`

D

`42`

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To solve the problem of finding the number of increasing functions from set A to set B, where \( A = \{a_1, a_2, a_3, a_4, a_5, a_6\} \) and \( B = \{1, 2, 3, \ldots, 9\} \), with the conditions that \( a_{i+1} > a_i \) and \( a_i \neq i \), we will follow these steps: ### Step 1: Understand the Constraints We need to find increasing functions, which means that each element in A must map to a unique element in B such that \( a_{i+1} > a_i \). Additionally, we cannot have \( a_i = i \) for any \( i \). ### Step 2: Determine Possible Values for \( a_1 \) Since \( a_1 \) cannot be 1 (because \( a_1 \neq 1 \)), the smallest value \( a_1 \) can take is 2. We will consider different cases for \( a_1 \). ### Step 3: Case Analysis 1. **Case 1: \( a_1 = 2 \)** - Possible values for \( a_2, a_3, a_4, a_5, a_6 \) are from the set \( \{3, 4, 5, 6, 7, 8, 9\} \) (7 choices). - We need to choose 5 elements from these 7. - The number of ways to choose is given by \( \binom{7}{5} = 21 \). 2. **Case 2: \( a_1 = 3 \)** - Possible values for \( a_2, a_3, a_4, a_5, a_6 \) are from the set \( \{4, 5, 6, 7, 8, 9\} \) (6 choices). - We need to choose 5 elements from these 6. - The number of ways to choose is given by \( \binom{6}{5} = 6 \). 3. **Case 3: \( a_1 = 4 \)** - Possible values for \( a_2, a_3, a_4, a_5, a_6 \) are from the set \( \{5, 6, 7, 8, 9\} \) (5 choices). - We need to choose 5 elements from these 5. - The number of ways to choose is given by \( \binom{5}{5} = 1 \). ### Step 4: Summing the Cases Now we will sum the number of ways from all cases: - From Case 1: 21 ways - From Case 2: 6 ways - From Case 3: 1 way Total number of increasing functions = \( 21 + 6 + 1 = 28 \). ### Final Answer The total number of increasing functions from set A to set B, under the given constraints, is **28**. ---
To solve the problem of finding the number of increasing functions from set A to set B, where \( A = \{a_1, a_2, a_3, a_4, a_5, a_6\} \) and \( B = \{1, 2, 3, \ldots, 9\} \), with the conditions that \( a_{i+1} > a_i \) and \( a_i \neq i \), we will follow these steps: ### Step 1: Understand the Constraints We need to find increasing functions, which means that each element in A must map to a unique element in B such that \( a_{i+1} > a_i \). Additionally, we cannot have \( a_i = i \) for any \( i \). ### Step 2: Determine Possible Values for \( a_1 \) Since \( a_1 \) cannot be 1 (because \( a_1 \neq 1 \)), the smallest value \( a_1 \) can take is 2. We will consider different cases for \( a_1 \). ...
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