Question

Suppose a lot of n objects having ${n_1}$ objects of one kind, ${n_2}$ objects are of second kind, ${n_3}$ objects of third kind,….., ${n_k}$ objects of ${k^{th}}$ kind satisfying the condition ${n_1} + {n_2}..... + {n_k} = n,$ then the number of possible arrangements/permutation of m objects out of this lot is the coefficient of ${x^m}$ in the expansion m!\prod \left\\{ {\sum\limits_{\lambda = 0}^{{a_1}} {\dfrac{{{x^\lambda }}}{{\lambda !}}} } \right\\}
The number of permutations of the letters of the word SURITI taken $4$ at a time is
A.$360$
B.$240$
C.$216$
D.$192$

Answer 1

P(n,r) = n!/(n-r)! is actually said to be the formula for permutation of n objects for r selection of objects. So we are using this equation to solve the problem in this question.

Complete answer:
Given word:SURITI
I=$2$ times
S,U,R,T=$1 -$ times
Case-I; All letters are different
Number of arrangement ,
=$^5{C_4} \times 4!$

$$= 120 \\\$$

Case-II; two are different and two are same kind
Number of arrangement=

$$\\\$$

$= 6 \times \dfrac{{24}}{2} = 72$
Total arrangement=$120 + 72 = 192$
So there are $192$ permutations of the letters in the word SURITI when taken $4$ at a time. So we found that Option D- $192$ is the correct answer.

Additional Information
A permutation of a set is a loosely defined arrangement of its members into a sequence or linear order, or a rearrangement of its elements if the set is already ordered. The act or method of changing the linear order of an ordered set is often referred to as "permutation."
Permutations are distinct from combinations, which are random choices of certain members of a set.
Essentially, A permutation is a method of arranging objects in a specific order. When working with permutation, it's important to think about both selection and arrangement. In a nutshell, ordering is critical in permutations. To put it another way, a permutation is an ordered mixture.

Note:
The elements in permutation must be grouped in a specific order, while in combination, the order of the elements does not matter.When working with permutation, it's important to think about both selection and arrangement. In a nutshell, ordering is critical in permutations. To put it another way, a permutation is an ordered mixture.