In a group of boys the number of arrangement of 4 boys is 12 times the number ofarrangements of 2 boys. The number of boys in the group is(a) 10 (b) 8 (c) 6 (d) none of these

Let's solve the problem step by step:

Step 1: Let's assume the number of boys in the group is "x".

Step 2: The number of arrangements of 4 boys can be calculated using the formula for permutations: P(n, r) = n! / (n - r)!, where n is the total number of boys and r is the number of boys being arranged.

So, the number of arrangements of 4 boys is P(x, 4) = x! / (x - 4)!

Step 3: The number of arrangements of 2 boys can be calculated using the same formula: P(n, r) = n! / (n - r)!, where n is the total number of boys and r is the number of boys being arranged.

So, the number of arrangements of 2 boys is P(x, 2) = x! / (x - 2)!

Step 4: According to the problem, the number of arrangements of 4 boys is 12 times the number of arrangements of 2 boys. Mathematically, we can write this as:

P(x, 4) = 12 * P(x, 2)

x! / (x - 4)! = 12 * (x! / (x - 2)!)

Step 5: Simplifying the equation:

(x! / (x - 4)!) = 12 * (x! / (x - 2)!)

Canceling out the common terms:

(x - 2)(x - 1) = 12

Expanding the equation:

x^2 - 3x + 2 = 12

Rearranging the equation:

x^2 - 3x - 10 = 0

Step 6: Factoring the quadratic equation:

(x - 5)(x + 2) = 0

Step 7: Solving for x:

x - 5 = 0 or x + 2 = 0

x = 5 or x = -2

Since the number of boys cannot be negative, we can conclude that the number of boys in the group is 5.

Therefore, the correct answer is (d) none of these.