Solve by finding a combinatorial argument

Probability Level 2

n C 0 + n C 1 + n C 2 + n C 3 + . . . + n C n − 1 + n C n = ? ^n C_0 + ^n C_1 + ^n C_2 + ^n C_3 + ... + ^n C_{n-1} + ^n C_n =?

1 1 2 n − 1 2^{n-1} 2 n 2^n 2 n + 1 2^{n+1}

Harshal Gajjar by Harshal Gajjar , 22, India

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2 solutions

Kay Xspre
Nov 1, 2015

Notice that it is the binomial coefficient of the expansion of ( x + y ) n (x+y)^n . Just substitute x = y = 1 x=y=1 , then all the coefficient will have the sum of 2 n 2^n

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Harshal Gajjar
Oct 31, 2015

Hint: Expression in the question represents all possible ways of selecting student(s) from a class having n n students.

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