SECTION 12.6 The Binomial Theorem 899 1 Evaluate ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j The symbol n j , ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ read “ n taken j at a time,” is defined next. COMMENT On a graphing utility, the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟⎟ n j may be denoted by the key nCr . ■ Figure 21 Proof ( ) ( ) ( ) ( ) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = = = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = − = − − = n n n n n n n n n n n n n n 0 ! 0! 0 ! ! 0! ! 1 1 1 1 ! 1! 1 ! ! 1 ! 1 ! 1 ! You are asked to prove the remaining two formulas in Problem 45. ■ DEFINITION ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j If j and n are integers with ≤ ≤ j n 0 , the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j is defined as ( ) ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = − n j n j n j ! ! ! (1) Evaluating ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j Find: (a) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 3 1 (b) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 4 2 (c) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 8 7 (d) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 65 15 Solution EXAMPLE 1 (a) ( ) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = = ⋅ ⋅ ⋅ = = 3 1 3! 1! 3 1 ! 3! 1! 2! 3 2 1 1 2 1 6 2 3 (b) ( ) ( )( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = = ⋅ ⋅ ⋅ ⋅ ⋅ = = 4 2 4! 2! 4 2 ! 4! 2! 2! 4 3 2 1 2 1 2 1 24 4 6 (c) ( ) ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − = = ⋅ ⋅ = = 8 7 8! 7! 8 7 ! 8! 7! 1! 8 7! 7! 1! 8 1 8 ↑ = ⋅ 8! 8 7! (d) Figure 21 shows the solution using a TI-84 Plus CE graphing calculator. So ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ≈ × 65 15 2.073746998 1014 Now Work PROBLEM 5 THEOREM Four useful formulas involving the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j are ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = n n n n n n n n • 0 1 • 1 • 1 • 1
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