diff --git a/mathematics/pascal_triangle.r b/mathematics/pascal_triangle.r
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+# pascal_triangle.r
+# Script for generating and working with Pascal's Triangle.
+# Provides:
+#   - pascal_triangle(n): Returns a list of n rows of Pascal's Triangle (input: integer n > 0; output: list of numeric vectors).
+#   - pascal_row(n): Returns the nth row as a numeric vector (input: integer n >= 0; output: numeric vector).
+#   - pascal_element(n, k): Returns the binomial coefficient C(n, k) (inputs: integers n >= 0, 0 <= k <= n; output: numeric value).
+#   - print_pascal_triangle(n, centered=TRUE): Prints the triangle formatted (input: integer n > 0, logical centered; output: printed output).
+# No external dependencies.
+
+#' Generate Pascal's Triangle up to n rows
+#' @param n Number of rows
+#' @return List of rows, each containing the triangle values
+pascal_triangle <- function(n) {
+  if (n <= 0) {
+    stop("Number of rows must be positive")
+  }
+  
+  triangle <- list()
+  
+  for (i in 1:n) {
+    row <- numeric(i)
+    row[1] <- 1
+    row[i] <- 1
+    
+    if (i > 2) {
+      for (j in 2:(i-1)) {
+        row[j] <- triangle[[i-1]][j-1] + triangle[[i-1]][j]
+      }
+    }
+    
+    triangle[[i]] <- row
+  }
+  
+  return(triangle)
+}
+
+#' Generate single row of Pascal's Triangle
+#' @param n Row number (0-indexed)
+#' @return Vector containing the nth row
+pascal_row <- function(n) {
+  if (n < 0) {
+    stop("Row number must be non-negative")
+  }
+  
+  row <- numeric(n + 1)
+  row[1] <- 1
+  
+  for (i in 1:n) {
+    row[i + 1] <- row[i] * (n - i + 1) / i
+  }
+  
+  return(row)
+}
+
+#' Get specific element from Pascal's Triangle
+#' @param n Row number (0-indexed)
+#' @param k Column number (0-indexed)
+#' @return Value at position (n, k) - binomial coefficient C(n, k)
+pascal_element <- function(n, k) {
+  if (n < 0 || k < 0) {
+    stop("Row and column numbers must be non-negative")
+  }
+  if (k > n) {
+    return(0)
+  }
+  
+  # Use symmetry: C(n, k) = C(n, n-k)
+  if (k > n - k) {
+    k <- n - k
+  }
+  
+  result <- 1
+  for (i in seq_len(k)) {
+    result <- result * (n - i + 1) / i
+  }
+  
+  return(result)
+}
+
+#' Print Pascal's Triangle in formatted style
+#' @param n Number of rows
+#' @param centered Whether to center the triangle (default: TRUE)
+print_pascal_triangle <- function(n, centered = TRUE) {
+  triangle <- pascal_triangle(n)
+  
+  if (centered) {
+    # Calculate maximum width for centering
+    max_row <- triangle[[n]]
+    max_width <- sum(nchar(as.character(max_row))) + length(max_row) - 1
+    
+    for (i in 1:n) {
+      row <- triangle[[i]]
+      row_str <- paste(row, collapse = " ")
+      row_width <- nchar(row_str)
+      padding <- (max_width - row_width) / 2
+      
+      cat(rep(" ", floor(padding)), row_str, "\n", sep = "")
+    }
+  } else {
+    for (i in 1:n) {
+      cat(paste(triangle[[i]], collapse = " "), "\n")
+    }
+  }
+}
+
+#' Get diagonal sums from Pascal's Triangle
+#' @param n Number of rows
+#' @return Vector of diagonal sums (Fibonacci sequence)
+pascal_diagonal_sums <- function(n) {
+  triangle <- pascal_triangle(n)
+  sums <- numeric(n)
+  
+  for (i in 1:n) {
+    diagonal_sum <- 0
+    row <- i
+    col <- 1
+    
+    while (row >= 1 && col <= length(triangle[[row]])) {
+      diagonal_sum <- diagonal_sum + triangle[[row]][col]
+      row <- row - 1
+      col <- col + 1
+    }
+    
+    sums[i] <- diagonal_sum
+  }
+  
+  return(sums)
+}
+
+#' Get row sum of Pascal's Triangle
+#' @param n Row number (0-indexed)
+#' @return Sum of row n (equals 2^n)
+pascal_row_sum <- function(n) {
+  if (n < 0) {
+    stop("Row number must be non-negative")
+  }
+  return(2^n)
+    stop("Number of rows must be positive")
+
+#' Get all row sums up to n rows
+#' @param n Number of rows
+#' @return Vector of row sums
+pascal_all_row_sums <- function(n) {
+  if (n <= 0) {
+    return(numeric(0))
+  }
+  return(2^(0:(n-1)))
+}
+
+#' Generate Pascal's Triangle as matrix (padded with zeros)
+#' @param n Number of rows
+#' @return Matrix representation of Pascal's Triangle
+pascal_triangle_matrix <- function(n) {
+  triangle <- pascal_triangle(n)
+  mat <- matrix(0, nrow = n, ncol = n)
+  
+  for (i in 1:n) {
+    for (j in 1:length(triangle[[i]])) {
+      mat[i, j] <- triangle[[i]][j]
+    }
+  }
+  
+  return(mat)
+}
+
+#' Get odd numbers pattern in Pascal's Triangle (Sierpinski pattern)
+#' @param n Number of rows
+#' @return Matrix with 1 for odd, 0 for even
+pascal_odd_pattern <- function(n) {
+  triangle <- pascal_triangle(n)
+  mat <- matrix(0, nrow = n, ncol = n)
+  
+  for (i in 1:n) {
+    for (j in 1:length(triangle[[i]])) {
+      mat[i, j] <- triangle[[i]][j] %% 2
+    }
+  }
+  
+  return(mat)
+}
+
+# Examples
+if (FALSE) {
+  # Example 1: Generate and print Pascal's Triangle
+  cat("Example 1: Pascal's Triangle (10 rows)\n")
+  cat("======================================\n\n")
+  print_pascal_triangle(10)
+  
+  # Example 2: Generate specific row
+  cat("\nExample 2: Row 7 of Pascal's Triangle\n")
+  cat("======================================\n")
+  cat(sprintf("C(10, 3) = %.0f\n", pascal_element(10, 3)))
+  cat(sprintf("C(8, 4) = %.0f\n", pascal_element(8, 4)))
+  cat(sprintf("C(6, 2) = %.0f\n", pascal_element(6, 2)))
+  
+  # Example 3: Get specific element
+  cat("\nExample 3: Specific Elements\n")
+  cat("============================\n")
+  cat(sprintf("C(10, 3) = %d\n", pascal_element(10, 3)))
+  cat(sprintf("C(8, 4) = %d\n", pascal_element(8, 4)))
+  cat(sprintf("C(6, 2) = %d\n", pascal_element(6, 2)))
+  
+  # Example 4: Row sums (powers of 2)
+  cat("\nExample 4: Row Sums\n")
+  cat("===================\n")
+  sums <- pascal_all_row_sums(8)
+  for (i in 1:8) {
+    cat(sprintf("Row %d sum: %d = 2^%d\n", i-1, sums[i], i-1))
+  }
+  
+  # Example 5: Diagonal sums (Fibonacci sequence)
+  cat("\nExample 5: Diagonal Sums (Fibonacci Sequence)\n")
+  cat("=============================================\n")
+  diag_sums <- pascal_diagonal_sums(12)
+  cat("Diagonal sums:", paste(diag_sums, collapse = ", "), "\n")
+  cat("This is the Fibonacci sequence!\n")
+  
+  # Example 6: Pascal's Triangle as matrix
+  cat("\nExample 6: Pascal's Triangle as Matrix\n")
+  cat("======================================\n")
+  mat <- pascal_triangle_matrix(8)
+  print(mat)
+  
+  # Example 7: Odd/Even pattern (Sierpinski Triangle)
+  cat("\nExample 7: Odd Numbers Pattern (Sierpinski Triangle)\n")
+  cat("===================================================\n")
+  odd_pattern <- pascal_odd_pattern(16)
+  
+  for (i in 1:16) {
+    # Center the pattern
+    padding <- 16 - i
+    cat(rep(" ", padding))
+    for (j in 1:i) {
+      if (odd_pattern[i, j] == 1) {
+        cat("* ")
+      } else {
+        cat("  ")
+      }
+    }
+    cat("\n")
+  }
+  
+  # Example 8: Properties of Pascal's Triangle
+  cat("\nExample 8: Properties of Pascal's Triangle\n")
+  cat("==========================================\n")
+  
+  n <- 10
+  triangle <- pascal_triangle(n)
+  
+  cat("\n1. Symmetry Property:\n")
+  cat("Row 6:", paste(triangle[[7]], collapse = " "), "\n")
+  cat("(Each row is symmetric)\n")
+  
+  cat("\n2. Sum of row n = 2^n:\n")
+  for (i in 1:5) {
+    row_sum <- sum(triangle[[i]])
+    cat(sprintf("Row %d: sum = %d = 2^%d\n", i-1, row_sum, i-1))
+  }
+  
+  cat("\n3. Binomial Theorem:\n")
+  cat("(a + b)^4 = ")
+  row4 <- triangle[[5]]
+  terms <- character(5)
+  for (i in 1:5) {
+    power_a <- 4 - (i - 1)
+    power_b <- i - 1
+    coef <- row4[i]
+    
+    if (power_a == 0) {
+      terms[i] <- sprintf("%db^%d", coef, power_b)
+    } else if (power_b == 0) {
+      terms[i] <- sprintf("%da^%d", coef, power_a)
+    } else if (power_a == 1 && power_b == 1) {
+      terms[i] <- sprintf("%dab", coef)
+    } else if (power_a == 1) {
+      terms[i] <- sprintf("%dab^%d", coef, power_b)
+    } else if (power_b == 1) {
+      terms[i] <- sprintf("%da^%db", coef, power_a)
+    } else {
+      terms[i] <- sprintf("%da^%db^%d", coef, power_a, power_b)
+    }
+  }
+  cat(paste(terms, collapse = " + "), "\n")
+  
+  cat("\n4. Hockey Stick Pattern:\n")
+  cat("Sum of diagonal: 1 + 4 + 10 + 20 = 35 = C(7,3)\n")
+  cat("Formula: C(n,r) + C(n+1,r) + ... + C(n+k,r) = C(n+k+1,r+1)\n")
+  
+  cat("\n5. Catalan Numbers:\n")
+  # Catalan numbers: C(2n, n)/(n+1) for n = 1..4
+  catalan <- function(n) choose(2*n, n)/(n+1)
+  n_vals <- 1:4
+  cat("Catalan numbers for n=1..4:", paste(catalan(n_vals), collapse = ", "), "\n")
+  
+  # Example 9: Application - Probability
+  cat("\nExample 9: Application - Coin Flip Probability\n")
+  cat("===============================================\n")
+  cat("Flipping 5 coins, probability of exactly k heads:\n\n")
+  row5 <- pascal_row(5)
+  total <- sum(row5)
+  
+  for (k in 0:5) {
+    prob <- row5[k + 1] / total
+    cat(sprintf("%d heads: %d/%d = %.4f (%.2f%%)\n", 
+                k, row5[k + 1], total, prob, prob * 100))
+  }
+  
+  # Example 10: Large row computation
+  cat("Last 5 elements:", paste(row50[47:51], collapse = ", "), "\n")
+  cat("==================================\n")
+  cat("Row 50 (too large to display fully):\n")
+  row50 <- pascal_row(50)
+  cat("Number of elements:", length(row50), "\n")
+  cat("First 5 elements:", paste(row50[1:5], collapse = ", "), "\n")
+  cat("Middle element C(50,25):", format(row50[26], scientific = FALSE), "\n")
+  cat("Last 5 elements:", paste(row50[46:51], collapse = ", "), "\n")
+  
+  cat(sprintf("C(%d,%d) using Pascal's Triangle: %.0f\n", n, k, element))
+  cat(sprintf("C(%d,%d) using factorials: %.0f\n", n, k, factorial_calc))
+  cat("========================\n")
+  n <- 10
+  k <- 4
+  element <- pascal_element(n, k)
+  factorial_calc <- factorial(n) / (factorial(k) * factorial(n - k))
+  cat(sprintf("C(%d,%d) using Pascal's Triangle: %d\n", n, k, element))
+  cat(sprintf("C(%d,%d) using factorials: %d\n", n, k, factorial_calc))
+  cat(sprintf("Match: %s\n", ifelse(element == factorial_calc, "✓", "✗")))
+}