The Coefficient of X5 in the Expansion of ( 1 + X ) 21 + ( 1 + X ) 22 + . . . + ( 1 + X ) 30(A) 51c5 (B) 9c5 (C) 31c6 − 21c6 (D) 30c5 + 20c5 - Mathematics

MCQ

The coefficient of x5 in the expansion of $\left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . + \left( 1 + x \right)^{30}$

• 51C5

•  9C5

•  31C6 − 21C6

•  30C5 + 20C5

Solution

31C6 − 21C6

$\text{ We have } \left( 1 + x \right)^{21} + \left( 1 + x \right)^{22} + . . . \left( 1 + x \right)^{30}$
$= \left( 1 + x \right)^{21} \left[ \frac{\left( 1 + x \right)^{10} - 1}{\left( 1 + x \right) - 1} \right]$
$= \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$\text{ Coefficient of } x^5 \text{ in the given expansion = Coefficient of } x^5 \text{ in } \frac{1}{x}\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$= \text{ Coefficient of } x^6 \text{ in }\left[ \left( 1 + x \right)^{31} - \left( 1 + x \right)^{21} \right]$
$=^{31} C_6 -^{21} C_6$

Is there an error in this question or solution?

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 18 Binomial Theorem
Q 25 | Page 48
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.shaalaa.com/question-bank-solutions/the-coefficient-x5-expansion-1-x-21-1-x-22-1-x-30-a-51c5-b-9c5-c-31c6-21c6-d-30c5-20c5-introduction-binomial-theorem_55083", "fetch_time": 1619060366000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2021-17/segments/1618039560245.87/warc/CC-MAIN-20210422013104-20210422043104-00237.warc.gz", "warc_record_offset": 1068451332, "warc_record_length": 9659, "token_count": 514, "char_count": 1121, "score": 4.4375, "int_score": 4, "crawl": "CC-MAIN-2021-17", "snapshot_type": "latest", "language": "en", "language_score": 0.3839913308620453, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00037-of-00064.parquet"}