Roorkee University has to send 10 professors to 5 centers for its entrance examination, 2 to each center. Two of the enters are in Roorkee and the others are outside. Two of the professors prefer to work in Roorkee while three prefer to work outside. In how many ways can this be made if the preferences are to be satisfied?

(i) A person went to a restaurant for dinner. In the menu card, the person saw 10 Indian and 7 Chinese food items. In how many was the person can select either an Indian or a Chinese food? (ii) There are 3 typesof toy car and 2 types of toy train available in a shop. Find the number of ways a baby can buy a toy car and a toy train? (iii) How many of two-digit numbers can be formed using 1,2,3,4,5 without repetiton of digits? (iv) There persons enter in to a conference hall in which there are 10 seats, In how many ways they can take their place? (v) In how many ways 5 persons can be seated in a row?

(i) A kabaddi coach has 14 players ready to play. How many different teams of 7 players could the coach put on the court? (ii) There are 15 persons in a party and if , each 2 of them shakes hands wit each with each other. How many handshakes happen in the party? (iii) How many chords can be drawn through 20 points on a circle? (iv) In a parking lot one hunderd, one year old cars are parked. Out of them five are to be chosen at random for to check its pollution devices. How many different set of five cars are possible? (v) How many ways can a team of 3 boys, 2 girls and 1 transgender be selected from 5 boys, 4 girls and 2 transgender?

A person always prefers to eat parantha and vegetable dish in his meal. How many ways can he makes his platter in a marriage party if there are three types of paranthas, four types of vegetable dish, three types of salads and two types of sauces? a. 3360 b. 4096 c. 3000 d. none of these

There are three routes R_1,R_2and R_3 from Madhu's home to her place of work. There are four parking lots P_1, P_2, P_3, P_4 and three entrances B_1, B_2, B_3 into the office building. There are two elevatos E_1 and E_2 to her floor. Using the tree diagram explain how many ways can she reach her office?

A company manufactures two proudcts. A and B on which the profits erarned per unit areRs. 30 and Rs. 40 respectively. Each product is procced on two machines M_(1) and M_(2) . One unit of product A requires on hour of processing time on M_(1) and 2 hours on M_(2) While one unit of product B requires one hour each on M_(1) and M_(2) . Machines M_(1) and M_(2) are availabe atmost 10 hours and 12 hours respectivrely during working day. The company wants to know how many units of proudcts A and B should be produced to maximise the profit. to formulate a linearprogramming problem (i) Write the objective function. (ii) Write all contraints.

a. Determine the electrostatic potential energy of a system consisting of two charges 7 muC and -2muC (and with no external field) placed at (-9 cm, 0,0) and (9 cm, 0, 0) respectively. b. How much work is required to separate the two charges infinitely away from each other? c. Suppose that the same system of charges is now placed in an external electric field E=A (1/r^(2)) , A =9 xx 10^(5) Cm^(-2) . What would the electrostatic energy of the configuration be?