A) \[^{55}{{C}_{4}}\]
B) \[^{55}{{C}_{3}}\]
C) \[^{56}{{C}_{3}}\]
D) \[^{56}{{C}_{4}}\]
Correct Answer: D
Solution :
\[^{50}{{C}_{4}}{{+}^{55}}{{C}_{3}}{{+}^{54}}{{C}_{3}}{{+}^{53}}{{C}_{3}}{{+}^{52}}{{C}_{3}}\] \[{{+}^{51}}{{C}_{3}}{{+}^{50}}{{C}_{3}}\] \[{{=}^{50}}{{C}_{3}}{{+}^{50}}{{C}_{4}}{{+}^{51}}{{C}_{3}}{{+}^{52}}{{C}_{3}}{{+}^{53}}{{C}_{3}}{{+}^{54}}{{C}_{3}}\] \[{{+}^{55}}{{C}_{3}}\] \[{{=}^{51}}{{C}_{4}}{{+}^{51}}{{C}_{3}}{{+}^{52}}{{C}_{3}}{{+}^{53}}{{C}_{3}}{{+}^{54}}{{C}_{3}}{{+}^{55}}{{C}_{3}}\] \[{{=}^{52}}{{C}_{4}}{{+}^{52}}{{C}_{3}}{{+}^{53}}{{C}_{3}}{{+}^{54}}{{C}_{3}}{{+}^{55}}{{C}_{3}}\] \[{{=}^{53}}{{C}_{4}}{{+}^{53}}{{C}_{3}}{{+}^{54}}{{C}_{3}}{{+}^{55}}{{C}_{3}}\] \[{{=}^{54}}{{C}_{4}}{{+}^{54}}{{C}_{3}}{{+}^{55}}{{C}_{3}}\] \[{{=}^{55}}{{C}_{4}}{{+}^{55}}{{C}_{3}}\] \[{{=}^{56}}{{C}_{4}}\]
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