The number of terms in the multinomial expansion of (a + b + c + d)¹⁰ in which the exponents of a, b, c , and d are all mutually distinct is

Step-by-step solution

The general term in the expansion of (a + b + c + d)¹⁰ is of the form 10! x! y! z! w! a^x b^y c^z d^w , where x, y, z, w are non-negative integers such that x + y + z + w = 10 . We are required to find the number of terms where x, y, z, w are mutually distinct. Let us list the possible partitions of 10 into 4 distinct non-negative integers. Assuming x If x = 0 , the remaining sum is 10 . The distinct partitions into three positive integers are: 1 + 2 + 7 = 10 1 + 3 + 6 = 10 1 + 4 + 5 = 10 2 + 3 + 5 = 10 This gives