How many 'words' can be formed using all of the letters of BANANA if each
1 point word must start with the letter $A$ ?
Answer
Final Answer: The number of 'words' that can be formed using all of the letters of BANANA, starting with the letter 'A', is \(\boxed{30}\).
Step 1 ：The word 'BANANA' has 6 letters, with 3 'A's, 2 'N's and 1 'B'. We are asked to find the number of words that can be formed using all of these letters, but with the condition that the word must start with 'A'.
Step 2 ：This is a permutation problem with repetition. The formula for permutations with repetition is \(\frac{n!}{n1! * n2! * ... * nk!}\) where \(n\) is the total number of items, and \(n1, n2, ..., nk\) are the number of each type of item.
Step 3 ：In this case, we have \(n = 5\) (since one 'A' is already used at the start), \(n1 = 2\) (for 'A'), \(n2 = 2\) (for 'N') and \(n3 = 1\) (for 'B').
Step 4 ：So, we need to calculate \(\frac{5!}{2! * 2! * 1!}\).
Step 5 ：By calculating, we get \(\frac{120}{4} = 30\).
Step 6 ：Final Answer: The number of 'words' that can be formed using all of the letters of BANANA, starting with the letter 'A', is \(\boxed{30}\).
