Question: A science administrator is reviewing grant applications and notices that each letter of a 7-character activity code is either 'E', 'X', or 'Y'. How many such codes contain exactly three 'E's and at least one pair of consecutive identical letters?

["Title: Counting Valid 7-Character Activity Codes: Exactly Three 'E's with Consecutive Identical Letters", "---", "A science administrator managing research grant applications encounters a pattern-based challenge: each valid 7-character activity code uses only the letters 'E', 'X', and 'Y', contains exactly three 'E's, and must include at least one pair of consecutive identical letters. How many such codes satisfy these criteria?", "In this SEO-optimized guide, we break down how to count these specific scientific identifier codes by analyzing letter frequency and applying combinatorial reasoning—perfect for data-driven grant management and algorithmic problem solving.", "---", "### Step 1: Total Codes with Exactly Three 'E's (No Restrictions)", "First, count all 7-letter codes using only 'E', 'X', 'Y' with exactly three 'E's. The remaining 4 positions are filled with either 'X' or 'Y'—so 2 choices per position.", "- Choose 3 positions out of 7 for the 'E's:\n [\n \binom{7}{3} = 35\n ]\n- For each such arrangement, the other 4 positions each have 2 options:\n [\n 2^4 = 16\n ]\n- Total unrestricted codes with exactly three 'E's:\n [\n 35 \ imes 16 = 560\n ]", "This total includes codes with and without consecutive identical letters. But the question requires at least one pair of consecutive identical letters—specifically, letters that are the same and appear side-by-side.", "---", "### Step 2: Subtract Codes with No Consecutive Identical Letters", "To find codes that do have at least one pair of consecutive identical letters, subtract those codes where no two adjacent letters are the same from the total.", "So, we now compute the number of 7-character codes with:\n- Exactly three 'E's\n- The remaining four positions filled with 'X' and 'Y' (each position ∈ {X, Y})\n- No two consecutive letters are identical", "This is a constrained permutation problem with strict adjacency restrictions.", "---", "### Step 3: Count Valid Codes with No Consecutive Identical Letters", "Let’s model building a 7-character string using 'E', 'X', 'Y', with:\n- Exactly three 'E's\n- The rest chosen from {'X', 'Y'}, with no two adjacent letters equal", "We must place three 'E's in such a way that no two 'E's are adjacent only if the non-'E' letters between them prevent repetition—but more critically, no two identical letters are adjacent, regardless of what they are.", "Because of the “no consecutive identical letters” rule, adjacent letters must alternate.", "Let’s define the structure: We are selecting positions for 3 'E's and 4 non-'E' letters (each X or Y), arranged so no two adjacent characters are equal.", "To count such sequences, we consider the full string as a sequence of 7 positions, placing the three 'E's such that:\n- They are not adjacent (to avoid consecutive duplicates), but even if not adjacent, adjacent non-'E's must differ.", "However, since non-'E' letters are only 'X' and 'Y', the real challenge lies in arranging the 4 non-'E' characters (each X or Y) such that no two are the same in succession, while placing the three 'E's in valid gaps.", "But a simpler and more effective approach is:", "We fix the pattern of letter types (not individual characters) that satisfies:\n- Length 7\n- Exactly 3 'E's\n- No two adjacent characters equal\n- The remaining 4 positions use only 'X' and 'Y'", "Let’s denote a valid letter type pattern: a sequence of length 7 over {E, X, Y} with exactly three E’s and four letters from {X, Y}, such that no two adjacent letters are identical.", "Let’s define such sequences recursively or via constructive counting.", "But due to the complexity, we use structured enumeration based on block patterns.", "---", "### Step 4: Use Inclusion with Valid Configurations", "Let us count the number of 7-character strings with:\n- Exactly 3 'E's\n- The other 4 positions filled with 'X' or 'Y' (2 choices each)\n- No two adjacent characters are equal", "We proceed by generating all valid sequences (E, X, Y pattern) meeting the criteria.", "Let’s denote:\n- The number of E's is fixed: 3\n- Non-E characters: 4, each X or Y → 2⁴ = 16 decisions\n- But not all 16 × 35 = 560 sequences avoid consecutive duplicates.", "Let’s use a known combinatorial insight: the number of t-letter sequences over a 3-letter alphabet with fixed letter counts and no adjacent duplicates.", "We use dynamic programming logic adapted for exact counts.", "Let’s define ( f(n, e, last) ) as the number of valid sequences of length ( n ), with exactly ( e ) 'E's used, and last character being ( last \in {E, X, Y} ), with no two adjacent equal.", "But for precision and competitive clarity, we estimate valid sequences via constructive counting.", "---", "### Step 5: Construct Valid Patterns with 3 E’s and No Consecutive Identicals", "Let’s count how many 7-letter strings with exactly 3 'E's and the other 4 letters chosen from {X, Y}, such that no two adjacent characters are equal.", "We break down by the placement of E's such that placing them doesn’t force identical adjacent non-'E' characters.", "Let’s list possible placements of the 3 'E's in 7 positions, with no two adjacent E's (strongly helpful), but carefully because non-'E' adjacent letters must differ too.", "Case A: E’s are not adjacent\nTo prevent consecutive E’s: no two E’s adjacent → treat as placing 3 non-adjacent E’s in 7 slots.", "This is equivalent to placing 3 E’s with at least one non-E between any two.", "Use the stars-and-bars method: number of ways to place 3 non-adjacent E’s in 7 positions is ( \binom{7 - 3 + 1}{3} = \binom{5}{3} = 10 )", "But this only ensures no E–E adjacency. Non-E adjacent characters must also differ.", "Now, in each such E placement, fill the 4 non-E positions with X/Y such that no two adjacent non-E letters are the same.", "But since non-E positions are not necessarily contiguous, we must analyze the full string pattern.", "Instead, we shift perspective: since the non-E positions form 4 isolated slots (not adjacent to each other or E’s), but may be adjacent to E’s—however, since E ≠ X and E ≠ Y, a non-E letter adjacent to an E is always different, so only adjacent non-E positions must strictly alternate.", "Therefore, the 4 non-E positions form independent or nearly independent blocks depending on E placement.", "But to simplify, note: if two non-E positions are adjacent in the sequence, they must be assigned different letters (X/Y)—but since they each have 2 choices, and can differ, the only restriction is dissimilarity when consecutive.", "Hence, to count sequences with no two identical adjacent letters, we require:\n- No E–E adjacency (a stronger condition than just non-consecutive E’s)\n- No X–X or Y–Y adjacency at positions where both are non-E", "But since E precedes or follows a non-E, the danger is:\n- Two adjacent non-E positions assigned same letter → invalid\n- E next to non-E → safe, since E ≠ X, E ≠ Y", "Thus, only consecutive non-E positions matter. So, we can model this as: for each valid placement of 3 non-adjacent E’s (to avoid E–E), assign X/Y to the 4 non-E positions such that every pair of consecutive non-E positions are assigned different letters—but only if those two are adjacent.", "But non-E positions may be separated.", "Let’s define: choose 3 positions for E from 7, such that no two are adjacent — this ensures no E–E adjacency.", "Number of ways to choose 3 non-adjacent positions in 7:\nThis is standard:\nLet ( a_n(k) ) = number of ways to choose k non-adjacent positions in n spots.", "Formula: ( \binom{n - k + 1}{k} ) →\n[\n\binom{7 - 3 + 1}{3} = \binom{5}{3} = 10\n]", "So, 10 valid placements where E’s are not adjacent.", "Now, in each such placement, there are 4 non-E positions. These positions may be clustered or spread.", "But as long as"]