' 9) How many strings of length 10 over the alphabet {a, b, c, d}…

Question

Manu · Accepted Answer

Step 1: Question 9: The Correct option is (C) Problem Statement We need to find the number of strings of length 10 using the alphabet { 𝑎 , 𝑏 , 𝑐 , 𝑑 } { a , b , c , d } that have exactly 3 a's. Step-by-Step Solution Choosing Positions for 'a' : We need to select 3 positions out of the 10 for the 'a' characters. The number of ways to do this is given by the binomial coefficient: ( 10 3 ) = 10 ! 3 ! ( 10 − 3 ) ! = 10 ! 3 ! ⋅ 7 ! ( 3 10 ​ ) = 3 ! ( 10 − 3 )! 10 ! ​ = 3 ! ⋅ 7 ! 10 ! ​ Calculating this: ( 10 3 ) = 10 × 9 × 8 3 × 2 × 1 = 120 ( 3 10 ​ ) = 3 × 2 × 1 10 × 9 × 8 ​ = 120 Filling Remaining Positions : The remaining 7 positions can be filled with any of the remaining letters { 𝑏 , 𝑐 , 𝑑 } { b , c , d } . Each position has 3 choices (b, c, or d). The number of ways to fill these 7 positions is: 3 7 3 7 Calculating this: 3 7 = 2187 3 7 = 2187 Combining Both Parts : The total number of strings is the product of the number of ways to choose positions for 'a' and the number of ways to fill the remaining positions: ( 10 3 ) × 3 7 ( 3 10 ​ ) × 3 7 Substituting the calculated values: 120 × 2187 = 262440 120 × 2187 = 262440 Explanation: The total number of strings of length 10 with exactly 3 a's is: ( 10 3 ) ⋅ 3 7 = 120 × 2187 = 262440 ( 3 10 ​ ) ⋅ 3 7 = 120 × 2187 = 262440 Step by step how we calculate and get 262440 Calculate the binomial coefficient ( 10 3 ) ( 3 10 ​ ) : ( 10 3 ) = 10 ! 3 ! ( 10 − 3 ) ! = 10 ! 3 ! ⋅ 7 ! ( 3 10 ​ ) = 3 ! ( 10 − 3 )! 10 ! ​ = 3 ! ⋅ 7 ! 10 ! ​ Calculate 10 ! 10 ! , 3 ! 3 ! , and 7 ! 7 ! : 10 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3 , 628 , 800 10 ! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3 , 628 , 800 3 ! = 3 × 2 × 1 = 6 3 ! = 3 × 2 × 1 = 6 7 ! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5 , 040 7 ! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5 , 040 Substitute these values into the binomial coefficient formula: ( 10 3 ) = 3 , 628 , 800 6 ⋅ 5 , 040 = 3 , 628 , 800 30 , 240 = 120 ( 3 10 ​ ) = 6 ⋅ 5 , 040 3 , 628 , 800 ​ = 30 , 240 3 , 628 , 800 ​ = 120 Calculate 3 7 3 7 : 3 7 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 2 , 187 3 7 = 3 × 3 × 3 × 3 × 3 × 3 × 3 = 2 , 187 Multiply the results: ( 10 3 ) ⋅ 3 7 = 120 × 2 , 187 = 262 , 440 ( 3 10 ​ ) ⋅ 3 7 = 120 × 2 , 187 = 262 , 440 So, the solution to the expression ( 10 3 ) ⋅ 3 7 ( 3 10 ​ ) ⋅ 3 7 is: 262 , 440 262 , 440 The correct answer is the third option: ( 10 3 ) ⋅ 3 7 ( 3 10 ​ ) ⋅ 3 7 Step 2: To determine the number of strings of length 12 over the alphabet { 𝑎 , 𝑏 , 𝑐 , 𝑑 } { a , b , c , d } that start with "aa" or end with "aa" or both, we can use the principle of inclusion-exclusion. Let's define: 𝐴 A : The set of strings that start with "aa". 𝐵 B : The set of strings that end with "aa". We are interested in finding ∣ 𝐴 ∪ 𝐵 ∣ ∣ A ∪ B ∣ , the number of strings that start with "aa" or end with "aa" or both. Using the principle of inclusion-exclusion, we have: ∣ 𝐴 ∪ 𝐵 ∣ = ∣ 𝐴 ∣ + ∣ 𝐵 ∣ − ∣ 𝐴 ∩ 𝐵 ∣ ∣ A ∪ B ∣ = ∣ A ∣ + ∣ B ∣ − ∣ A ∩ B ∣ First, let's find ∣ 𝐴 ∣ ∣ A ∣ , the number of strings that start with "aa": The first two positions are fixed as "aa". The remaining 10 positions can be any of the 4 letters (a, b, c, d). So, ∣ 𝐴 ∣ = 4 10 ∣ A ∣ = 4 10 . Next, let's find ∣ 𝐵 ∣ ∣ B ∣ , the number of strings that end with "aa": The last two positions are fixed as "aa". The remaining 10 positions can be any of the 4 letters (a, b, c, d). So, ∣ 𝐵 ∣ = 4 10 ∣ B ∣ = 4 10 . Now, let's find ∣ 𝐴 ∩ 𝐵 ∣ ∣ A ∩ B ∣ , the number of strings that start and end with "aa": The first two positions are fixed as "aa". The last two positions are fixed as "aa". The remaining 8 positions can be any of the 4 letters (a, b, c, d). So, ∣ 𝐴 ∩ 𝐵 ∣ = 4 8 ∣ A ∩ B ∣ = 4 8 . Now, we can calculate ∣ 𝐴 ∪ 𝐵 ∣ ∣ A ∪ B ∣ : ∣ 𝐴 ∪ 𝐵 ∣ = ∣ 𝐴 ∣ + ∣ 𝐵 ∣ − ∣ 𝐴 ∩ 𝐵 ∣ ∣ A ∪ B ∣ = ∣ A ∣ + ∣ B ∣ − ∣ A ∩ B ∣ ∣ 𝐴 ∪ 𝐵 ∣ = 4 10 + 4 10 − 4 8 ∣ A ∪ B ∣ = 4 10 + 4 10 − 4 8 Let's calculate the values: 4 10 = 1048576 4 10 = 1048576 4 8 = 65536 4 8 = 65536 So: ∣ 𝐴 ∪ 𝐵 ∣ = 1048576 + 1048576 − 65536 ∣ A ∪ B ∣ = 1048576 + 1048576 − 65536 ∣ 𝐴 ∪ 𝐵 ∣ = 2097152 − 65536 ∣ A ∪ B ∣ = 2097152 − 65536 ∣ 𝐴 ∪ 𝐵 ∣ = 2031616 ∣ A ∪ B ∣ = 2031616 Explanation: Therefore, there are 2,031,616 strings of length 12 over the alphabet { 𝑎 , 𝑏 , 𝑐 , 𝑑 } { a , b , c , d } that start with "aa" or end with "aa" or both. Final Answer: 9. T he correct answer is the third option: ( 10 3 ) ⋅ 3 7 ( 3 10 ​ ) ⋅ 3 7 10. The number of strings of length 12 over the alphabet { 𝑎 , 𝑏 , 𝑐 , 𝑑 } { a , b , c , d } that start with "aa" or end with "aa" or both is 2 , 031 , 616 2 , 031 , 616

AI tools

Study help

Free tools

Earn & more

Computer science

Computer science

Manu

Answer

Extracted Transcript from Images

AI

Study help

Earn & more

Company