: Đa thức $P\left( x \right)={{\left( 1+3x+2{{x}^{2}} \right)}^{10}}={{a}_{0}}+{{a}_{1}}x+…+{{a}_{20}}{{x}^{20}}$. Tìm ${{a}_{15}}$

: Đa thức $P\left( x \right)={{\left( 1+3x+2{{x}^{2}} \right)}^{10}}={{a}_{0}}+{{a}_{1}}x+…+{{a}_{20}}{{x}^{20}}$. Tìm ${{a}_{15}}$

C. ${{a}_{15}}=C_{10}^{10}.C_{10}^{5}{{.3}^{5}}+C_{10}^{9}.C_{9}^{6}{{.3}^{3}}+C_{10}^{8}.C_{8}^{7}.3.$

B. ${{a}_{15}}=C_{10}^{10}.C_{10}^{5}{{.2}^{5}}+C_{10}^{9}.C_{9}^{6}{{.2}^{6}}+C_{10}^{8}.C_{8}^{7}{{.2}^{7}}$

C. ${{a}_{15}}=C_{10}^{10}.C_{10}^{5}{{.3}^{5}}{{.2}^{5}}+C_{10}^{9}.C_{9}^{6}{{.3}^{3}}{{.2}^{6}}+C_{10}^{8}.C_{8}^{7}{{.2}^{7}}$

D. ${{a}_{15}}=C_{10}^{10}.C_{10}^{5}{{.3}^{5}}{{.2}^{5}}+C_{10}^{9}.C_{9}^{6}{{.3}^{3}}{{.2}^{6}}+C_{10}^{8}.C_{8}^{7}{{.3.2}^{7}}$

Hướng dẫn

Chọn D

Ta có: $P\left( x \right)={{\left( 1+3x+2{{x}^{2}} \right)}^{10}}=\sum\limits_{k=0}^{10}{C_{10}^{k}}{{\left( 3x+2{{x}^{2}} \right)}^{k}}$

$=\sum\limits_{k=0}^{10}{C_{10}^{k}}\sum\limits_{i=0}^{k}{C_{k}^{i}}{{(3x)}^{k-i}}.{{(2{{x}^{2}})}^{i}}=\sum\limits_{k=0}^{10}{C_{10}^{k}}\sum\limits_{i=0}^{k}{C_{k}^{i}}{{.3}^{k-i}}{{.2}^{i}}{{x}^{k+i}}$

với $0\le i\le k\le 10\,\,$. Do đó $k+i=15$ với các trường hợp

$k=10,i=5$ hoặc $k=9,i=6$ hoặc $k=8,i=7$

Vậy ${{a}_{15}}=C_{10}^{10}.C_{10}^{5}{{.3}^{5}}{{.2}^{5}}+C_{10}^{9}.C_{9}^{6}{{.3}^{3}}{{.2}^{6}}+C_{10}^{8}.C_{8}^{7}{{.3.2}^{7}}$.