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شائع الجبر >

expand (x+y)^{23}

الحلّ

expand

(x + y)^{23}

الحلّ

x^{23} + 23 x^{22} y + 253 x^{21} y^{2} + 1771 x^{20} y^{3} + 8855 x^{19} y^{4} + 33649 x^{18} y^{5} + 100947 x^{17} y^{6} + 245157 x^{16} y^{7} + 490314 x^{15} y^{8} + 817190 x^{14} y^{9} + 1144066 x^{13} y^{10} + 1352078 x^{12} y^{11} + 1352078 x^{11} y^{12} + 1144066 x^{10} y^{13} + 817190 x^{9} y^{14} + 490314 x^{8} y^{15} + 245157 x^{7} y^{16} + 100947 x^{6} y^{17} + 33649 x^{5} y^{18} + 8855 x^{4} y^{19} + 1771 x^{3} y^{20} + 253 x^{2} y^{21} + 23 x y^{22} + y^{23}

خطوات الحلّ

(x + y)^{23}

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

:فعّل نظريّة ذات الحدّين لنيوتن

a = x, b = y

= i = 0 \sum 23 (i 23) x^{(23 - i)} y^{i}

وسّع حاصل الجمع

= \frac{23 !}{0 ! ( 23 - 0 ) !} x^{23} y^{0} + \frac{23 !}{1 ! ( 23 - 1 ) !} x^{22} y^{1} + \frac{23 !}{2 ! ( 23 - 2 ) !} x^{21} y^{2} + \frac{23 !}{3 ! ( 23 - 3 ) !} x^{20} y^{3} + \frac{23 !}{4 ! ( 23 - 4 ) !} x^{19} y^{4} + \frac{23 !}{5 ! ( 23 - 5 ) !} x^{18} y^{5} + \frac{23 !}{6 ! ( 23 - 6 ) !} x^{17} y^{6} + \frac{23 !}{7 ! ( 23 - 7 ) !} x^{16} y^{7} + \frac{23 !}{8 ! ( 23 - 8 ) !} x^{15} y^{8} + \frac{23 !}{9 ! ( 23 - 9 ) !} x^{14} y^{9} + \frac{23 !}{10 ! ( 23 - 10 ) !} x^{13} y^{10} + \frac{23 !}{11 ! ( 23 - 11 ) !} x^{12} y^{11} + \frac{23 !}{12 ! ( 23 - 12 ) !} x^{11} y^{12} + \frac{23 !}{13 ! ( 23 - 13 ) !} x^{10} y^{13} + \frac{23 !}{14 ! ( 23 - 14 ) !} x^{9} y^{14} + \frac{23 !}{15 ! ( 23 - 15 ) !} x^{8} y^{15} + \frac{23 !}{16 ! ( 23 - 16 ) !} x^{7} y^{16} + \frac{23 !}{17 ! ( 23 - 17 ) !} x^{6} y^{17} + \frac{23 !}{18 ! ( 23 - 18 ) !} x^{5} y^{18} + \frac{23 !}{19 ! ( 23 - 19 ) !} x^{4} y^{19} + \frac{23 !}{20 ! ( 23 - 20 ) !} x^{3} y^{20} + \frac{23 !}{21 ! ( 23 - 21 ) !} x^{2} y^{21} + \frac{23 !}{22 ! ( 23 - 22 ) !} x^{1} y^{22} + \frac{23 !}{23 ! ( 23 - 23 ) !} x^{0} y^{23}

\frac{23 !}{0 ! ( 23 - 0 ) !} x^{23} y^{0}

بسّط:

x^{23}

\frac{23 !}{1 ! ( 23 - 1 ) !} x^{22} y^{1}

بسّط:

23 x^{22} y

\frac{23 !}{2 ! ( 23 - 2 ) !} x^{21} y^{2}

بسّط:

253 x^{21} y^{2}

\frac{23 !}{3 ! ( 23 - 3 ) !} x^{20} y^{3}

بسّط:

1771 x^{20} y^{3}

\frac{23 !}{4 ! ( 23 - 4 ) !} x^{19} y^{4}

بسّط:

8855 x^{19} y^{4}

\frac{23 !}{5 ! ( 23 - 5 ) !} x^{18} y^{5}

بسّط:

33649 x^{18} y^{5}

\frac{23 !}{6 ! ( 23 - 6 ) !} x^{17} y^{6}

بسّط:

100947 x^{17} y^{6}

\frac{23 !}{7 ! ( 23 - 7 ) !} x^{16} y^{7}

بسّط:

245157 x^{16} y^{7}

\frac{23 !}{8 ! ( 23 - 8 ) !} x^{15} y^{8}

بسّط:

490314 x^{15} y^{8}

\frac{23 !}{9 ! ( 23 - 9 ) !} x^{14} y^{9}

بسّط:

817190 x^{14} y^{9}

\frac{23 !}{10 ! ( 23 - 10 ) !} x^{13} y^{10}

بسّط:

1144066 x^{13} y^{10}

\frac{23 !}{11 ! ( 23 - 11 ) !} x^{12} y^{11}

بسّط:

1352078 x^{12} y^{11}

\frac{23 !}{12 ! ( 23 - 12 ) !} x^{11} y^{12}

بسّط:

1352078 x^{11} y^{12}

\frac{23 !}{13 ! ( 23 - 13 ) !} x^{10} y^{13}

بسّط:

1144066 x^{10} y^{13}

\frac{23 !}{14 ! ( 23 - 14 ) !} x^{9} y^{14}

بسّط:

817190 x^{9} y^{14}

\frac{23 !}{15 ! ( 23 - 15 ) !} x^{8} y^{15}

بسّط:

490314 x^{8} y^{15}

\frac{23 !}{16 ! ( 23 - 16 ) !} x^{7} y^{16}

بسّط:

245157 x^{7} y^{16}

\frac{23 !}{17 ! ( 23 - 17 ) !} x^{6} y^{17}

بسّط:

100947 x^{6} y^{17}

\frac{23 !}{18 ! ( 23 - 18 ) !} x^{5} y^{18}

بسّط:

33649 x^{5} y^{18}

\frac{23 !}{19 ! ( 23 - 19 ) !} x^{4} y^{19}

بسّط:

8855 x^{4} y^{19}

\frac{23 !}{20 ! ( 23 - 20 ) !} x^{3} y^{20}

بسّط:

1771 x^{3} y^{20}

\frac{23 !}{21 ! ( 23 - 21 ) !} x^{2} y^{21}

بسّط:

253 x^{2} y^{21}

\frac{23 !}{22 ! ( 23 - 22 ) !} x^{1} y^{22}

بسّط:

23 x y^{22}

\frac{23 !}{23 ! ( 23 - 23 ) !} x^{0} y^{23}

بسّط:

y^{23}

= x^{23} + 23 x^{22} y + 253 x^{21} y^{2} + 1771 x^{20} y^{3} + 8855 x^{19} y^{4} + 33649 x^{18} y^{5} + 100947 x^{17} y^{6} + 245157 x^{16} y^{7} + 490314 x^{15} y^{8} + 817190 x^{14} y^{9} + 1144066 x^{13} y^{10} + 1352078 x^{12} y^{11} + 1352078 x^{11} y^{12} + 1144066 x^{10} y^{13} + 817190 x^{9} y^{14} + 490314 x^{8} y^{15} + 245157 x^{7} y^{16} + 100947 x^{6} y^{17} + 33649 x^{5} y^{18} + 8855 x^{4} y^{19} + 1771 x^{3} y^{20} + 253 x^{2} y^{21} + 23 x y^{22} + y^{23}

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