Question

The number of combinations that can b formed with 5 oranges, 4 mangoes and 3 bananas when it is essential to take
(i) at least one fruit be” k”
(ii) one fruit of each kind “m”
Find 2m-k

Answer 1

In this question we have three kinds of fruit that are oranges, mangoes and bananas. We can solve this question by using the combination method. First, we have to find the value of k and m then we can put these values in the given equation.

Complete step-by-step answer:
We have given in the question
Oranges-5
Mangoes-4
Bananas-3
Here, 5 oranges are alike of one kind, 4 mangoes are alike of second kind and 3 bananas are alike of third kind
(i)the required number of combination (at least one fruit)
$= (5 + 1)(4 + 1)(3 + 1){2^0} - 1$
We know that ${2^0} = 1$, putting the value
$= 6.5.4 - 1$
Multiply the numbers and subtract the total by 1
$= 120 - 1$
Subtracting the numbers we get
=$119$
Here the value of k = 119
(ii) The number required for combination (one fruit of each kind)
$= {}^5{C_1}.{}^4{C_1}.{}^3{C_1}$
${}^n{C_r} = \dfrac{{n!}}{{r!(n - 1)!}}$
Solve the above combination using the formula
$= \dfrac{{5!}}{{1!(5 - 1)!}}.\dfrac{{4!}}{{1!(4 - 1)!}}.\dfrac{{3!}}{{1!(3 - 1)!}}$
$= \dfrac{{5.4.3.2.1}}{{4.3.2.1}}.\dfrac{{4.3.2.1}}{{3.2.1}}.\dfrac{{3.2.1}}{{2.1}}$
$= 5.4.3$
$= 60$
So, we have the value of m = 60
Putting the values of k and m in equation 2m-k
$= 2m - k$
$k = 119,m = 60$
$= 2.60 - 119$
$= 120 - 119$
$= 1$

Hence the answers are $k = 119,m = 60$ and $2m - k = 1$

Note: As questions contain conditions, we have to use a combination method to solve the question.do the question step by step for getting the correct answer. A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of selection does not matter. In combination, you can select the item of any order.