Chapter 12: Q.49 (page 837)

$(\begin{matrix} 5 \\ 0 \end{matrix}) {(\frac{1}{4})}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(\frac{1}{4})}^{4} (\frac{3}{4}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(\frac{1}{4})}^{3} {(\frac{3}{4})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(\frac{1}{4})}^{1} {(\frac{3}{4})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(\frac{3}{4})}^{5} = ?$

Short Answer

Expert verified

The value of $(\begin{matrix} 5 \\ 0 \end{matrix}) {(\frac{1}{4})}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(\frac{1}{4})}^{4} (\frac{3}{4}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(\frac{1}{4})}^{3} {(\frac{3}{4})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(\frac{1}{4})}^{1} {(\frac{3}{4})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(\frac{3}{4})}^{5} = 1$

Step by step solution

Step 1. Given information

Here

$(\begin{matrix} 5 \\ 0 \end{matrix}) {(\frac{1}{4})}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(\frac{1}{4})}^{4} (\frac{3}{4}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(\frac{1}{4})}^{3} {(\frac{3}{4})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(\frac{1}{4})}^{1} {(\frac{3}{4})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(\frac{3}{4})}^{5} i s g i v e n$

we have to find out the value of the given statement

Step 2. Description of finding the value of given question

By using the binomial theorem we can find the value for this problem.

Binomial Theorem states that

$L e t x a n d a b e r e a l n u m b e r s . F o r a n y p o s i t i v e i n t e g e r n, w e h a v e {(x + a)}^{n} = (\begin{matrix} n \\ 0 \end{matrix}) x^{n} + (\begin{matrix} n \\ 1 \end{matrix}) a x^{n - 1} + (\begin{matrix} n \\ 2 \end{matrix}) a^{2} x^{n - 2} + . . . . . + (\begin{matrix} n \\ n \end{matrix}) a^{n} T h e g i v e n q u e s i o n (\begin{matrix} 5 \\ 0 \end{matrix}) {(\frac{1}{4})}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(\frac{1}{4})}^{4} (\frac{3}{4}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(\frac{1}{4})}^{3} {(\frac{3}{4})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(\frac{1}{4})}^{1} {(\frac{3}{4})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(\frac{3}{4})}^{5} i s g i v e n i s i n t h e f o r m o f b i n o m i a l t h e o r e m e x p a n s i o n . S o c o m p a r e d t o e x p a n d e d f o r m w e g e t n = 5, x = (\frac{1}{4}) a n d a = (\frac{3}{4}) t h e n b y a p p l y i n g b i n o m i a l e x p a n s i o n f o r m w e g e t {(\frac{1}{4} + \frac{3}{4})}^{5} = {(\frac{4}{4})}^{5} = {(1)}^{5} = 1, s o v a l u e o f (\begin{matrix} 5 \\ 0 \end{matrix}) {(\frac{1}{4})}^{5} + (\begin{matrix} 5 \\ 1 \end{matrix}) {(\frac{1}{4})}^{4} (\frac{3}{4}) + (\begin{matrix} 5 \\ 2 \end{matrix}) {(\frac{1}{4})}^{3} {(\frac{3}{4})}^{2} + (\begin{matrix} 5 \\ 3 \end{matrix}) {(\frac{1}{4})}^{2} {(\frac{3}{4})}^{3} + (\begin{matrix} 5 \\ 4 \end{matrix}) {(\frac{1}{4})}^{1} {(\frac{3}{4})}^{4} + (\begin{matrix} 5 \\ 5 \end{matrix}) {(\frac{3}{4})}^{5} i s 1$

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Step 2. Description of finding the value of given question

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50145+5114434+52143342+53142343+54141344+55345=?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Description of finding the value of given question

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Decision Maths

Study anywhere. Anytime. Across all devices.