Explanation Given information: The expansion, ( 5 0 ) ( 1 4 ) 5 + ( 5 1 ) ( 1 4 ) 4 ( 3 4 ) + ( 5 2 ) ( 1 4 ) 3 ( 3 4 ) 2 + ( 5 3 ) ( 1 4 ) 2 ( 3 4 ) 3 + ( 5 4 ) ( 1 4 ) ( 3 4 ) 4 + ( 5 4 ) ( 3 4 ) 5 . Formula used: Binomial Theorem: Let x and y be real numbers. For any positive integer n , ( x + y ) n = ( n 0 ) x n + ( n 1 ) y x n − 1 + ...... + ( n j ) y j x n − j + .... + ( n n ) y n = ∑ j = 0 n ( n j ) y j x n − j Calculation: Now use binomial theorem, Comparing the expansion ( 5 0 ) ( 1 4 ) 5 + ( 5 1 ) ( 1 4 ) 4 ( 3 4 ) + ( 5 2 ) ( 1 4 ) 3 ( 3 4 ) 2 + ( 5 3 ) ( 1 4 ) 2 ( 3 4 ) 3 + ( 5 4 ) ( 1 4 ) ( 3 4 ) 4 + ( 5 4 ) ( 3 4 ) 5 with the general expansion ( n 0 ) x n + ( n 1 ) y x n − 1 + ...... + ( n j ) y j x n − j + .... + ( n n ) y n , the values are x = 1 4 , y = 3 4 , n = 5 . Now using binomial theorem, ( 5 0 ) ( 1 4 ) 5 + ( 5 1 ) ( 1 4 ) 4 ( 3 4 ) + ( 5 2 ) ( 1 4 ) 3 ( 3 4 ) 2 + ( 5 3 ) ( 1 4 ) 2 ( 3 4 ) 3 + ( 5 4 ) ( 1 4 ) ( 3 4 ) 4 + ( 5 4 ) ( 3 4 ) 5 = ( 1 4 + 3 4 ) 5 ⇒ ( 5 0 ) ( 1 4 ) 5 + ( 5 1 ) ( 1 4 ) 4 ( 3 4 ) + ( 5 2

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Chapter 12.5, Problem 49AYU

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To calculate: The value of (50)(14)5+(51)(14)4(34)+(52)(14)3(34)2+(53)(14)2(34)3+(54)(14)(34)4+(54)(34)5

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Chapter 12.5, Problem 48AYU

Chapter 12.5, Problem 50AYU

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Chapter 12 Solutions

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The value of ( 5 0 ) ( 1 4 ) 5 + ( 5 1 ) ( 1 4 ) 4 ( 3 4 ) + ( 5 2 ) ( 1 4 ) 3 ( 3 4 ) 2 + ( 5 3 ) ( 1 4 ) 2 ( 3 4 ) 3 + ( 5 4 ) ( 1 4 ) ( 3 4 ) 4 + ( 5 4 ) ( 3 4 ) 5

Solution Summary: The author explains how to calculate the value of (c5 0end

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To calculate: The value of (50)(14)5+(51)(14)4(34)+(52)(14)3(34)2+(53)(14)2(34)3+(54)(14)(34)4+(54)(34)5

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